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preparer Tits sip titth past a dal atatet ga 


ay ha ealoya) ate Perr tree Tt 
pisiss Roep hint waked moa pmelt operand vate siebetatsiehstssebed tte 






















aban ina page 
Te cgi 
sel mit nas anes Hy bd gti 
Plebrg rept ata vee 
signagenetaijoopensiinh 
ins Pyey meek opab egy ob 


peptone tie waders beef 



































Aled Abas ah enenellel 


















































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gah 9s oh (rel seen ate Phar Preorrenene tit Nohara rec an si} onset eines thirsty tatebani st eres ha 

. bieikse ate v od Oc bbe game siaie paar aces aed al eheiall regia tind ae Hat robes siehetal 

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sixia jae cade etl gh arnt erenerela hm ahaa magn tae agade Saar ately ea ee iM 

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ach hipaa vpresahyea waked ats 10a) al ay ageing entatriaiaiat aja ty ayes pee teh ye apae 

: Hisi pi wiageeat ase hag Beene nite aaeopblneseinerets 4) aie baie oheray f Cevhibt heh pratt wie pale trite et tieah 
vt 





Nein tyne) etal hehe begat ote taleds peteday® jerniahai aa 4) miei rotten aebetedera ay adie phe. 

ona Perry Siabededag sneadobalandeh whrtad steven ene sears ae ihe eit ‘ i siento 

Ageanee Unive ti ry sce Sahopanal athe ecane taigi are boa lribeba tere tas Shatitel hazed ras miyeh sal #4 
re ey 


a 2 sitalea attates: 
iene aitalea wtgisoh ebuita ly jylapiabatia isin ee earhaane 











‘ Stabe ga eke 
: tieeteddtseatatonsneted sesh: THR eather Hysheratateng paige sian sadaeye 
awe | nie te tachees tala say aeadadetan teat setaggradedyibedet silvnsh of rd ebaaedey ny ates eee t abaya aaa 
cnehadaiue® agate i abose aie andt Lape nery tere 
sip miabe jmcadepetedaiue ree? i chee slaraia photsi 
EYP +y se 









ati eaipr et 

jad en neg s jay ee eve) 74 setelana inst iveye 
haghek ja pk, otha ang pee sjaapaiagoeneat ‘a 
A ae resedosateheven, a ahe eee 4 NS 


oa ee poring tabreeradyirnte eiedettn\ at edoys 

+o 5 ake ere Pebers wee ye te) Hail + ole | ' 

venga aise egg ieee CO) aie) gator piphaks bud oe aoa cieecbded ty: i 
error irrerrirr iCal ry. et hias gia albpperpae’ ebatece boy al eetahaila ‘ali ae 





















at ae cat aati he 





145) Te Hi 


















































4) ae 9 
POP eT gl ator EA otal rhs bien dr et eF sa hallinta ethn'd ob hgh st by Tt 
te fataiet all nye Ad day eietaba] (16het bur ‘ icaitapetseet vate (ha 
hayes hee ey eaibed se geienke) tiv Labatt \ sateen ay oe j 
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fede eahed sakemiebshes fe eddacahalseee ihe ahaa abhi ht Hie isnt srbhs ryt 
maerenee terrier enw retort pon) Ped dapat aaa: ‘ Siaealal ia i Petir irs rane 
jar heist ema i hein’ t 
ca ha hotasa tay eyed tual Me crea ciated afeh 








Aveo Siege 62 = 
“ 5 ay Berane reeq eyPnet sas? 
5 aye fas Fhe hs Pee Oe ee Ga ie OEP 494 
ca wig in goa > elma [aed rae 
ap latay gh ahenpeet ett & 
lage eae ae aeene eG8 aie 
sneha it yeh ‘ pues 
Tk Be did e, @) Age, save Mb 
se rhat waguate) abets 
ae yw late 
yaerdeir 


eee dacs apet ad aay STL bh bahaeeaiyy hades 
9 ets! o pert he abatepoorn 4) 46H $e 
wen for ego) 08s fm ts “ie 
cUTper eb eda sh, ne * ahaa Peer Pee ere Leth 
wepeiicind ated ia Hig mailas Bat aihye npaitade iy 
aes AN aid gts f2ty dhereinhel si ge ded as si geri the 


adda ha, whates WP as? ale 
suing, wha} a1 ves) xh 





bce}somrs aba fet 
feet eys 















fete laden aah euaa te ji 



















say ate eee a 
fo 












rani? 





ori 















soraparehpishavny 
ahve tha its addi 
at fx eb a a He bcallet sie bepnyined 
te eahat eanleaqaente(edalle sail alhad a: 4 
# He Tae 2 vag 
vray He aieateita ne reat 
wa shay haiate ok 

me sri gatk isectitet ‘aii iinet 
gounay lla pailava seta latte 
jadebatey nh aaitaidtath ies sétaih 















Aad ath aj stl 
SP ANBE Of d Ede dich 
14 Sha}, phuket 























diab iat a7 we | ei 
t siete iy 
paling a) de a 
(gata alana a 


hogs ie meieiviviss eeeb® 
poate gaa bapa pat eG? 








aet by 19 fa 


ey ee 





ee ba ss rari 
Lait 



























ted a 
paged ath ahwd leselata’ told aie a 


pia 
sae Shes pth a-arball qi) 2} qibanial wibatty ethae saa 
dai abvnsionwladatagahal #7) awtall #7 sidghauaaladbpvantgenchia iets 


if apa 
Simoabagadauaegia Fpgateieeeyy Spal titoka ji Olam alin td beled? tage Taiiatels! 
PUreL ida: ttathiis i-aiinahtat ieee rota fas patie ae Sask Sisley 
bets eeu pisseesiat jad | Sarde ahi af mieted pst sibys tit bd ail tahabe 
sd 446 of aie ab oa apaees eye rer eee 
pred she de yl ay spain emia aR ails pena 
ah es tha pat aia ca aaah 
sh dtsaageradeheda TResed ff 
Jailed assed rah eH ot 
He tha yeaptaleityts 
. ote i djayeda pei alah - 
shang siedaggi sian eid 
ajo pajalaase Weare! teat th 
tabeesaadal Mh steel iba teal 
fade alhayha nail ri a ai 
ieee pete Retntaaegat me st atane 
Lajadufig satis aganadanaiate! iat he 
erin 

















pai siaae 
404401059 Hi rhapeieiohgt¢ 
aeoialanahahabatnliietsl segs 
2 15 re fab ah aes) a 088 abi h- bed 

ora yal-atapenaial al iid bt 
apeoatetae 






























Dhl 
ie 
tinea aeh 
4 








sagiviaiaoegaiee 
photena laterals ris 
pha la dete gate intirhe 
“ine fobaial sisiatatne 
[cla tetie leteia aia ba faiaastte iam t¥) . 
ajeserslist iafal ace ah abiialsts ee 
iwiattaee dare, andes, 
‘aad Pere a ee Was 
cw tiailat a canaom iar g 4 1e jebedemly de 
rehwde cudaelion + jaliae ay fy ate! ary dais 
: Aavevane Liha} jabstaanads 
ese! wi sinh sl viapene: 2) hea 
selata soe tel eiete paledatteedet steadied nf fl 
ya savane soma tad ahiont siz mingle alee saya) hy 
payadrbagel alae ralleth ref 




































v4! TREE 
a tia ebs at net gles : : 

pai qe i isigals qhtat ey wid salad tases 
Sita a aca Hien agaialay tyay 
read igeeyahy jwigvdieth edie 4 a) 
oreaa) a ae 







































Mi 
‘eieieit 


tebe ih petty 

se 
erry 1 
Nit iat at aii 
Ray as 



























































at tein 











jana sen bey a (eae aites 
viata tapenalmie tarate 












parapa yey tt 
wiatahaly jalape\ era ieee agtatadigy Seis bari Mantis is: rary ie 
ahalatese eharsiiejet Fee epaien® Ld nit Gee @ el dehs 
Pe et a oneredacaat fae ananeieie pada shale via oats Treatenitvore mi Heiavetstat 
fad ghedaanleg tata tag ats dat adie ie iva anda tthe 





a) 4)ai2 Fah gpalegriatvngl? trid pahasggaset ei ete) Paid & 


cHereremeaprpiprtc strana. ivr ttc) Lite aes bikin 
atat apates ¢ 


a bhatt al} tia ayaa onan galrie 


4 feat eteves phetes 
ate bata talla ty 


jae faseisidte) ae 




























































ie} Sie hg ad aie FompN Na Fed etree 
yee baie the e/a pe ie Mele yah ale baht ona ook the iahekenal glad daha talet jalapata tema satats se tenaie dl ah) ’ ‘ 
ee siadeleiadeia sx aienenabatega axaeel aid HiFi e 34 alg idle) ate Ordebrhacaagal a! sigh als rns pba ue hd aia iedmiet Hite aad ee Hye ity 
Per CUERPO Or TL 1 eure rastancnes Dek heat Kab ahal pds wisiwtatalataoniahe cereal Manieibetnesa dia eka anes sitet rate ai } 
Liottnde GaP gue Epay pe Preemie ee RPT TT OT arse Gr Ree hr te amt sabre gierevedargy agus hyeeyd bpycticrenymebnane pat Aj aialeda tenants hatte Hinge pert 
Ty lee Sah nhaare a ialatoneae ce wiahans) 1 004i inl by fw ss wits iynaiatalinleed joi sit jeitrsatail aul tt ay siiayaa eldata sitet ooh Ft ita nana 
Te ele rar cies tategate) eaiaisje le lnrekapererat ah ait Wi Ala Lgaa ebb bake cadt Satins r vPialeta taal salotaanhatets tah) iat ‘ ried it realy 4 
aac ahaa ws nal 4 wate! bid ketene pital disdape rang sae maa 40a) Lhe Aiaieteay javatMvtat sled nlata ats ais ( 
pds Ok rege ge alee Ca ba ee mb eae sare Tt} Hist sidiaiad bagias aa gaas arate iat ielael either ahi taseie ed id lah thine yasiainat via ivias et ts . 
eee by wept tebmi-peath abe PIAS itataie oahatseatetebt taney ehaueabal lai gona Ly: shel ek taehatesadelgseaaes 
chess jain) agent eater ieee bias 4!) AW taal aati ab ailafal shany panier ba *thandia pale ‘ i 
Test hatte iatateg e popaibed aban tia Sayed dat dna tae nah atti 4 ine iM 





rnin a yahIN 8 er ahe! oh wes apege aes! * 
ee erie am * 
Pepe! eayeye Tet 
ote cis rcasieye piaieie trie intzati? f 
reg arepheds mesial Mien eee 
Laitesa a led ejaledipty ht} Me waagsh Sa Loeb 


oh<t Sn gusarapete sa [sent 
eta aah 


Haile ja tal etedakanheba heals 1e estate) 
vituds ial stabi add pb at iy ait 
joints toate Reda eee sual maint 
nha it 





tiiendislate ba rerabey 
tet ey aah dtegedad ss eso qos Aiea 
abassba jabs lane sts ett avert 
Lobgtag weet sqhviplia pana jae fase haat 

Patalerabe 
vig dedi ates 

























Aas 
site 
Haigh altetang faite’ 4 
¢ tw lates qa saaiairnadatal vital 
jobs colstaicGshitetsl sit Saiore ot 
telah ole phadetalatapeiapel alans 

eadehsiei Haletateeatattelahabedanel aie jal waa 
heRd sabe la abel bagedeyeietets cab ihaiaiey) atiat 
4ighelal hele ay siege@! erro he vere te ul Jala ija atiasten whey} as wae Hreet Hh 
ai dda js dda da eh dba! wide tenets bahay al aneaiald dca alah spar enact ater riser { ane dae al 

apphayala Hae ghguade ee aah fey jabs diay glateudbe sald de ieitee Leta tad aid (Aad 8iep oa Peal an4 
ped said ta Pur ieleiead bet cig ARAB Fhanatab dy facie mtane daeliehaieisataiad sss iano a aha 

Sabaiaiese sifia'ha 0g) |aapa tar) ae talent’ shit , 

pha latalie peers pr teAD 
Yeh ofiy radiata 



















tin de wish rade slats iter 

a aleyaye otacel 

y lap an agi cele 
gieies 4 





gi Bw) ath y lade 















fealty iid na @iahe ete tey 
Ate tighvhs Parsi yee 
fad aay sities 







































‘ eet 
faieganunine 


(hehe thal dtetnaahelatgataite 


hoe ad 































iaeaty hiro shape 
a) ate a fetane atleleiedeiet 


Chale twtattejene ehecgia at 
haat ste lsieie tale 


aie bp deiabe' ala panera res! 
gehs atyhielaies alivehe ave 
iz Lallatibaah ods! pyaido ade 
dra iaadabah aad aha 








tiene iets 


eapeey 
















eee ie Cal aaa) herve ei ; 
» < eT did ford pooner alt girasane ya iat ¥ fe t 
se riencmristetate aceie eedtonser igine guarentee ‘ site disdana srt 
gaiage ste seine ibiniaiene } thi Halelsia hnaanah it 
realy vif rine 






+4 pipe teipe 

Unbeorernys ir iT) ate ig ian SHisk 
aap jeral either tay add eapatis ae 
Siaihei a tala a telatel ai sMete th 
palin etalay adananelale aie 

Jeraglaidiet 

di ajah via jek sbecwsate aisle a ah aRalts 
ab ahe'nay spe beie eG ibe heee No éhejala rity 
shee gal bdslatapan apabsitt jel tt ahalpa peas i 


Yea Had ato Lad etaileta ei at ih othal) afl: fe tay 
ia taper lat ahaseuel one ae ! af ha jako a haba li futnlea haied alba sho 
i + thbagesala toda ite: 


aves te tabcrs bye i i}: 

haba widtalieds hast ane Sa laiahulabatenalgiatadatabatia pane = 
Saye heratatetie ts nislaialgl ae padatatatate ais be ue orth aati 
Hetqia ee shadstiala late aia folie ha touan lal) a peated a ba ” ry sal 


alta te bol ne 





18 woo oh 


wee gab aiale's wel a tsi asia 









! (aba! inh 
4 7 pat 

oate ids esa finan eta aiaiale 
ait wih aha aala bed abs a|A)8t4 Ureted ingeg 

ura dssailetatenatanel rk Dyed ty a 
Hato Mantra ta 


patialabale 
5 i ich Taathedaiieleda iv jis ba atta eral se 
ip ree Ma co se 









piarsiia adeta st 34 jegeditp eine ye }4 fate 
S grabeneliertia's jai atete 
ede dy repatotel ages dst dhaladela el 
arr dia stg Waal hale 
ope Hla te Se) sale lelete 

tala ial eiaia 

















abel eles 


ieise 


bapaiiainna rathendie tates wi atte 


























ais F es i} 























wleieiete 
(albeit anata lo! Aye 
vieiataletebe aia iad 



























mele jalegaedn egies i pubtey eb sealers 
















yo pepadeliekaiohey ate! ns 
4 5 te ha wen tachi Le tt Shah w igs giitle lat itl ays aif atta to haul ed sh aitatnhe arbh) a nashaya ie! 
ie” ae e's 4 ahanetete “ aS plortnyaceeg ate resetelala ehaieid ale let siiaie siete tia tha be ta ha deepen 
‘sane s gals a oat dace ranean (0s api eoblaad gana) tae Fy ete" sash ale i eioge se jed c MarR pair ore 
Deine te) s rate ls foie ciate 
He oie tetarelsaidieig iat tei oe 
ronreraih aja taniaiedet ae 


7 . 
1 dk eke 28 2 eeheted pin Vietisiaiglesg a al waite 
Shae Sage, pe borne saat ye ¥ 
(agua leq wags barf ates oie oe 
. 


= ter wins aeseites wh 
diebasatsnaiananayat shard 





























nt aharnatias 

























































we PEM PaND ete Bie )a #2 8 
si whconbrain yO NG) tree ete 4 ere dated sab sitet 
ba ee ee le aly fetate tenes peprbeve teed ne Prt Py) aaa) Lab shee 
la a bay 2 hehe fabs ipat oe ew le dis esueaiat dirtanee Haare ¢ anges phaladatedalatalhelalaitaatas ditstaless ‘ ae 
a) ald daha be sec at reelet eat 1aw cat rhe lets lap stented Pal se iiiay atats(. jars abaya eevee letshedc vate eSarenens 
4p baat) ahaye patadaletg MiG 959 fat Via baloiabe anti OW aie aheal “ehg arte aed sha haede ah ailakerates ane eet eee gh be sdepel ¢ 
ja latehe tase harete bea ang dads! ate vba eda hula nates sedaiasdtatabeyatatac sures adatarte' y 
lode danntabata ds tansione ibsii chal atohatena tataleid tela: Heat FY 





Back viete teboeehit om 
ca ietaed go) ea Ee fat iat be 


gaane ata 


thats ne 








shaiijabite! peal fatagy deanatad pala 





ri 
oh 














oath nei ane yigs ry ent Hag alia a e 
aie pana sabe gal al at wih af aba faa edt So sdagtatisi| aia ii - 

Weyer suena y Pyialete a il ala anelte fl tha'td balgedeell tact baie 

diye tpeailedeibalie® vaitate bY tide 

Singhs tab wieiaiba 





F shale ha aia adebalee 
ot Leiba yeaa te 
9 aia eided tah thee 

bor Hives anetqieemte 
ohe habatininae ati al aah ob 2 44 qgiehe het rtdhe 
Te landy anges baat atattena te peverdhett 


Jato igransive tatel eke aie 
lenetons at 


a ahakese Sal shate de) qlay 
ha (fine pana) shade tal Pa aa 
alae deetdldiady niet oade 


apie 


iaseneey 
radars 
aie fetaiate wie ont 
hehe nasel Sadehe yet ehacata' 
THbaghde ied te rept ¥y aeetete 
is atelensishly 
sp acaseletote® esata tedeibatere 
avirera.) 4 haiabe: 
ha lg Selote gad para aet (alien et sateha seiatiave eb ole uate 
te! ‘ q aa plea tata) HN alate ate pign it shabbat . 
at as 
a8 eispriageda eernto Lait laiegstscte'é ps tanet Miata otetal tes pre 
shabamaddrdituraoa esa taleteite 
anaes ai ciandiais ; rhaiat efediys ig leiaie Reprint iss 
TaMebecinie! atgletal ahd td aitaue 
lena hal alate apni Hrprte tes Agbehaie) 
Collate enh teta daeaigda pelstead bole ae 
frahad ailella telly (gihgeg edhe! Dae asataeatete 
‘Libs ba manshha sa dtodd 40086 ‘he hy he 





veloari4 oie whe) 
gig atepel dng 

















a) sd) ehedenete gata 
Phy ee dtu latgha 

















diprpeted 
veer tr tee 


fefttiter te 
sajna raver 
“he 





















‘he he 
states dastateys 
te haengneeat 


bie bg be aa deja tet tie fe'eiee 
te fate halreaa theohaher’ te atetone heen sted 


stshoeulsiald 2 









shalabera telieiche aster Pts \e 

















re lew aga gia 
a jets beet Mohs sg bok aihote te Latee 
paiaia ahem igh adele a graeme, gaa) aaah 
selks tetera che eee aa 
eieta ane ion aahad 

hal § 










ogee ‘a oho d-wbene 
+ jg sk te fap he ae! Bone 


wines a Vey 












ate aataiga! 
wypeha keels piaerey ate 
wa labad Hi in 








y wralarerhsye 
aaa! jake ee a tebe 


5 pi ahs netateee eed pe caval a4 





THE UNIVERSITY | 
OF ILLINOIS 
LIBRARY 


ae fe 


DIAN: 





MATHEMATIOS 


Return this book on or before the 
Latest Date stamped below. A 
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L161—H41-- 








| THE TEACHING 


OF 


ELEMENTARY MATHEMATICS 


BY 


Bir 
THT BERS eo 
Dep, ATIOS 


BEG £m we, > 
is BY ERY 


DAVID EUGENE SMITH 


PRINCIPAL OF THE STATE NORMAL SCHOOL AT BROCKPORT 
NEW YORK 


Nef Work 


| THE MACMILLAN COMPANY 
LONDON: MACMILLAN & CO., Lt. 
1QOI 


All rights reserved 








CopyRIGHT, 1900, 


By THE MACMILLAN COMPANY. 





Set up and electrotyped February, 1900. Reprinted January, 
Igol. 


Nortooot JAress 
J.S. Cushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 





ALORS PRERLAGE 





Ir is evident that the problem of preparing a work 
upon the teaching of elementary mathematics may be 
attacked from any one of various standpoints. <A writer 
may confine himself to model lessons, for example; or 
to the explanation of the most difficult portions of the 
subject matter; or to the psychology of the subject; or 
to the comparison of historic methods; or to the exploit- 
ing of some hobby which he has ridden with success; or 
LO those devices which occupy so much time in the ordi- 
nary training of teachers. He may say, and with truth, 
that elementary mathematics now includes trigonom- 
try, analytic geometry, and the calculus; and that 
therefore a work with this title should cover the ground 
of Dauge’s “ Méthodologie,” or of Laisant’s masterly 
work, “ La Mathématique.” He may proceed dogmati- 
cally, and may lay down hard and fast rules for teaching, 
2xcusing this destruction of the teacher’s independence 
py the thought that the end justifies the means. But 
with a limited amount of space at his disposal, what- 
over point of attack he selects he must leave the 
thers more or less untouched; he cannot condense 

encyclopedia of the subject in three hundred pages. 


aie 
r 
wipes 
fie 
hatin 
Nees 


ry 
|) 


vi AUTHOR’S PREFACE 


Several years ago the author set about to find somk- ° 
thing of what the world had done in the way of making» 
and of teaching mathematics, and to know the really | 
valuable literature of the subject. He found, however, 
no manual to guide his reading, and so the accumulation 
of a library upon the teaching of the subject was a slow 
and often discouraging work. This little handbook is 
intended to help those who care to take a shorter, clearer 
route, and to know something of these great questions 
of teaching, — Whence came this subject? Why am I 
teaching it? How has it been taught? What should 
I read to prepare for my work? The subject is thus 
considered as in a state of evolution, while comparative 
method rather than dogmatic statement is the keynote. 
It is true that certain types are suggested, — methods, 
they are often called; but these are given as represent- 
ing the present development of the subject, and not as 
finalities. The effort has been, throughout, to set forth 
the subject as in a state of progress to which forward 
movement the teacher is to contribute; we have quite 
enough literature representing the static element. 

Considerable attention has been given to the bibliog 
raphy of the subject. At the risk of being accused oi 
going beyond the needs of teachers, the author has sugy. 
gested the most helpful works in French and Germany) 
as well as in English, and has not hesitated to quot 
from them. The body of the page is, however, alway 
in English, —the footnotes may be used or not, as t’ 














AUTHOR’S PREFACE Vil 





reader wishes. Where a quotation seemed to lose some- 
thing by being put into English, the original has been 
placed in a footnote. By these references the reader is 
put in touch with those works which the author has 
found of great value to him. The references might 
easily be multiplied, but this has not seemed desirable. 
There are many books on the teaching of mathematics, 
some of them quite pretentious in their claims, a few 
published in America, a few in England and France, 
and a large number in Germany. To cite all, or even 
a majority of these, might be positively harmful; it is 
hoped that the selection made has been reasonably 
judicious. 

If this work shall help, even in a small way, to open 
a wider field, or to offer a better point of view, to some- 
one just entering the profession, the author will feel 
repaid for his labors. 


DAVID EUGENE SMITH. 


STATE NORMAL SCHOOL, BROCKPORT, N.Y., 
January, 1900. 


a UE 


MODEORIS TIN TRODUGEION 


PERHAPS no single subject of elementary instruction 
has suffered so much from lack of scholarship on the 
part of those who teach it as mathematics. Arithmetic 
is universally taught in schools, but almost invariably 
as the art of mechanical computation only. The true 
significance and the symbolism of the processes em- 
ployed are concealed from pupil and teacher alike. 
This is the inevitable result of the teacher’s lack of 
mathematical scholarship. | 

The subtlety, delicacy, and accuracy of mathematical 
processes have the highest educational value, both 
direct and indirect. To treat them as mechanical rou- 
tine, not susceptible of explanation or illumination from 
a higher point of view, is to destroy in large measure 
the value of mathematics as an educational instrument, 
and to aid in arresting the mental development of the 
pupil. 

As long ago as the time of Aristotle it was pointed 
out that mathematics should not be defined in terms 
of the content with which it deals, but rather in terms 


ix 


x EDITOR’S INTRODUCTION 


of its method and degree of abstractness. Kant says/of 
mathematics, in the “ Critique of Pure Reason,” “'Bhe 
science of mathematics presents the most brilliant ex- 
ample of how pure reason may successfully enlarge 
its domain without the aid of experience.”! He then 
goes on to point out the ground of the distinction 
between philosophical and mathematical knowledge, 
and adds: ‘‘Those who thought they could distinguish 
philosophy from mathematics by saying that the former 
was concerned with gwality only, the latter with gwan- 
tity only, mistook effect for cause. It is owing to the 
form of mathematical knowledge that it can refer to 
guanta only, because it is only the concept of quantities 
that admits of construction, that is, of @ priori repre- 
sentation in intuition, while qualities cannot be repre- 
sented in any but empirical intuition.” ? 

Mr. Charles S. Peirce has recently made the criti- 
cism that Kant was not justified in supposing that 
mathematical and philosophical necessary reasoning 
are distinguished by the circumstance that the former 
uses construction or diagrams. Mr. Peirce holds that 
all necessary reasoning whatsoever proceeds by con- 
structions, and that we overlook the constructions in 
philosophy because they are so excessively simple. ? 
He goes on to show that mathematics studies nothing 
but pure hypotheses, and that it is the only science 


1 Miiller’s Translation (New York, 1896), p. 572.  ? Jdid.,, p. 573. 
3 Educational Review, 15, 214. 


Y EDITOR’S INTRODUCTION x1 


which never inquires what the actual facts are. It 
is “‘the science which draws necessary conclusions.” 

This acute argument is, I think, at fault in its con- 
FA sion that construction is employed in philosophical 
reasoning, but is otherwise sound. It fails, however, 
to point out clearly these facts :— 

1. The human mind is so constructed that it must 
see every perception in a time-relation —in an order — 
and every perception of an object in a space-relation 
—as outside or beside our perceiving selves. 

2. These necessary time-relations are reducible to 
Number, and they are studied in the theory of number, 
arithmetic and algebra. 

3. These necessary space-relations are reducible to 
Position and Form, and they are studied in geometry. 

Mathematics, therefore, studies an aspect of all 
knowing, and reveals to us the universe as it presents 
itself, in one form, to mind. To apprehend this and 
to be conversant with the higher developments of 
mathematical reasoning, are to have at hand the means 
of vitalizing all teaching of elementary mathematics. 

In the present - book, the purpose of which is_to 


present in ‘simple and succinct form to teachers the 
results of mathematical scholarship, to be absorbed by 
them and applied in their class-room teaching, the 
author has wisely combined the genetic and the ana- 
lytic methods. He shows how the elementary mathe- 
matics has developed in history, how it has been used 


xii EDITOR'S INTRODUCTION 


in education, and what its inner nature really is. It 
may safely be asserted that the elementary mathe- 
matics will take on a new reality for those who study 
this book and apply its teachings. 


NICHOLAS MURRAY BUTLER. 


COLUMBIA UNIVERSITY, NEW YORK, 
February 1, 1900. 


CONTENTS 


CHAPTER I 


PAGE 
HISTORICAL REASONS FOR TEACHING ARITHMETIC. — Impor- 
tance of the question. The evolution of reasons. The 
beginning utilitarian. Early correlation. Utilitarian among 
trading peoples. Tradition and examinations. The cul- 
ture value. As aremunerative trade. Asa mere show of 
knowledge. As an amusement. As a quickener of the 

wit. Scientific investigation of reasons . ; : I-18 


COAPTE RA 


Wuy ARITHMETIC IS TAUGHT AT PRESENT. — Two general 
reasons. The utility of arithmetic overrated. The culture 
value. Teachers generally fail here. Recognition of the 
culture value. What chapters bring out the culture value. 
What may well be omitted. Relative value of culture and 


utility ' : : : : ‘ ; ; : 19-41 


CHAPTER III 


How ARITHMETIC HAS DEVELOPED. — Reasons for studying 
the subject. Extent of the subject. The first step — 
counting. The second step—notation. The next great 
step in arithmetic. The twofold nature of ancient arith- 
metic. Arithmetic of the middle ages. The period of 
the Renaissance. Arithmetic since the Renaissance. The 
present status of school arithmetic . . : : 42-70 

Xll1 


Xiv CONTENTS 


CHAPTER IV | 
| PAGE 
How ARITHMETIC HAS BEEN TAUGHT. — The value of the’ 


investigation. The departure from object teaching. Rhym- 
ing arithmetics. Form instead of substance. Instruction, 
in method. Pestalozzi, Tillich. Reaction against Pesta- 
lozzi. Grube. Recent writers 71-97 


CHARTER. 


THE PRESENT TEACHING OF ARITHMETIC. — Objects aimed 
at. The number concept. The great question of method. 
The writing of numbers. The work of the first year. 
The time for beginning the study. Oral arithmetic. 
Treating the processes simultaneously. The spiral method. 
Common vs. decimal fractions. Improvements in algorism. 
The formal solution. Longitude and time. Ratio and 
proportion. Square root. The metric system. The ap- 
plied problems. Mensuration. Text-books. Explana- 
tions. Approximations. Reviews 98-144 


CHAPTER VI 


THE GROWTH OF ALGEBRA. — Egyptian algebra. Greek 
algebra. Oriental algebra. Sixteenth century algebra. 
Growth of symbolism. Number systems. Higher equa- 


tions 145-160 
CHAPTER VII 


\ ALGEBRA, WHAT AND WHY TAUGHT.— Algebra defined. The 
function. Why studied. Training in logic. Ethical 


value. When studied. Arrangement of text-books. 161-174 


CHAPTER. Vill 


TYPICAL PARTS OF ALGEBRA.— Outline. Definitions. The 
awakening of interest. Statinga problem. Signs of aggre- 
gation. Thenegativenumber. Checks. Factoring. The 


) CONTENTS XV 


PAGE 
emainder theorem. The quadratic equation. Equiva- 
lent equations. Extraneous roots. Simultaneous equa- 
tions and graphs. Methods of elimination. Complex 
numbers. The applied problems. The interpretation of 
solutions . : : ; 175-223 


CHAPTER IX 


THE GROWTH OF GEOMETRY. — Its historical position. The 
dawn of geometry. Geometry in Egypt; in Greece. Re- 
cent geometry. Non-Euclidean geometry ‘ - 224-233 


CHAPTER Sx. 


WHAT IS GEOMETRY? GENERAL SUGGESTIONS FOR TEACH- 
ING. — Geometry defined. Limits of plane geometry. 
The reasons for studying. Geometry in the lower grades. 
Intermediate grades. Demonstrative geometry. The use 
of text-books . : ; ; : ; : - 234-256 

cy, 


CHAPTER XI 
THE BASES OF GEOMETRY.— The bases. The definitions. 
Axioms and postulates. : é ; ‘ - 257-270 
CHAPTER XII 


TYPICAL PARTS OF GEOMETRY. — The introduction to demon- 
strative geometry. Symbols. Reciprocal theorems. Con- 
verse theorems. Generalization of figures. Loci of points. 
Methods of attack. Ratio and proportion. The impos- 
sible in geometry. Solid geometry . : ; - 271-296 


CHAPTER XIII 


THE TEACHER'S BOOK-SHELF. — Arithmetic. Algebra. Geom- 
etry. History and general method . ; ; » 297-305 


INDEX . ' : : ; é : ° é » 307-312 


< os 
é ‘Aaeaee < 


ser, ah 





Lda he 


PEeAGHING OF ELEMENTARY 
MATHEMATICS 


CHAR EERE 
HISTORICAL REASONS FOR TEACHING ARITHMETIC 


Importance of the question— For one who is pre- 
paring to teach any particular branch, and who hopes 
for success, the most important question is this: Why 
is the subject taught? More important than all meth- 
ods, more important than all devices or questions 
of text-books, or advice of the masters, is this far- 
reaching inquiry. Upon the answer depends the solu- 
tion of the problems relating to the presentation of the 
subject, the grade in which it should be begun, the 
time it should consume, the text-books, the methods, 
the devices,—-in fine, the general treatment of the 
whole matter in hand. It is the old, old cry, “We 
know not whither Thou goest, and how can we know 
the way?’ Unless the goal is known, what hope 
has one to find the path? 

Of course the inquiry is of no interest to the ma- 


chine teacher, the teacher who is content to follow 
B I 


2 THE TEACHING OF ELEMENTARY MATHEMATICS 


the book unthinkingly, to see the old curriculum re- 
main forever unchanged, and to follow the path his 
teacher trod, even though it be rough to the foot and 
without interest to the eye. But in England and 
America to-day we have a host of young and enthu- 
siastic teachers who are anxious to make the Anglo- 
Saxon educational system the best, and who are 
willing to inquire and to experiment. For such 
teachers this question is vital. 

The evolution of reasons— This search after reasons 
may be pursued either from the standpoint of a mere. 
inquirer into the conditions of to-day, or from that of 
one who is interested in the evolution of the ideas 
which are now in favor. While it is not possible in 
a work of this nature to enter into the details of the de- 
velopment of the reason for the presence of arithmetic 
in the curriculum to-day, some slight reference to this 
development may be of interest, and should be of value. 

The beginning utilitarian—In the far East, and 
in the far past, the reason for teaching arithmetic to 
children was almost always purely utilitarian. To the 
philosopher it was more than this, but in the early 
Chinese curricula it was given place merely that the 
boy might have sufficient knowledge of the four fun- 
damental processes for the common vocations of life.t 


1 Schmid, K. A., Geschichte der Erziehung vom Anfang an bis auf 
unsere Zeit, Stuttgart, 1884-98, Vol. I, p. 78. Hereafter referred to as 
Schmid, 


HISTORICAL FEASONS FOR TEACHING ARITHMETIC 3 


This was done in the common schools almost from 
the first, but in the middle ages! the subject so in- 
creased in importance that special schools were estab- 
lished for the study of arithmetic. A little later? it 
was taught as a special course in the high schools, 
open to those who had a taste in this direction, 
although even then children must have continued to 
learn common reckoning in the earlier years. In 
general, however, it has been taught in the far East 
for two thousand years, because of the utilities which 
it possesses, or merely for the purposes of examina- 
tion, or because it correlated with a study of the 
sacred books.? 

Early correlation — In India little could be expected 
for arithmetic in the schools. The aim of education, 
as summarized in the first book of Manu, was to bring 
man to lead a religious life. The reading of the Veda, 
the giving of alms, these were fundamental features of 
education.* Even to-day is this the case. For more 
than two thousand years the curriculum and the 
methods have remained quite unchanged, and even 
in our day, in the native schools, the boy’s work is 
largely that of memorizing the Hindu scriptures and 


1 Under the Sung dynasty, 961-1280. Schmid, I, p. 80. 

2 Under the Ming dynasty, 1368-1644. 

8 Laurie, S. S., Historical Survey of Pre-Christian Education, London, 
1895, p. 128, 141, 148. Hereafter referred to as Laurie. 

* Schmid, I, p, 105-107. 


4 THE TEACHING OF ELEMENTARY MATHEMATICS 


picking up other knowledge incidentally, a classical 
example of extreme correlation. For such people, 
arithmetic, beyond the mere rudiments, is of value 
only as it throws light upon the central subject, and 
hence it has little place in the curriculum.! 

The same idea characterized the early Mohammedan 
schools, where the Koran furnished the core of instruc- 
tion, a plan of education still obtaining, on a slightly 
more liberal scale, in the present schools of Islam.? It 
also held quite general sway in the monastic schools 
of the middle ages, where arithmetic, like everything 
else, was either warped to correlate with theology, or 
confined to the simplest calculations.? That arithmetic 
was popularly considered merely as having some slight 
value in trade is shown by a familiar bit of monkish 
doggerel, as old at least as the beginning of the fifteenth 
century.* It thus sets forth the values of the seven 
liberal arts, — grammar, dialectic, rhetoric, music, arith- 
metic, geometry, and astronomy: 


“Gramm. loquitur, Dia. vera docet, Rhe. verba colorat ; 
Mus. canit, Ar. numerat, Ge. ponderat, As. colit astra.” 


1 For a description of the arithmetic in the native Hindu schools of the 
present consult Delbos, L., Les Mathématiques aux Indes Orientales, Paris, 
1892, — pamphlet. 

2 Schmid, II (1), p. 599. 

Ib., II (1), p. 86. In this line is the rule attributed to Pachomius, 
“Omnino nullus erit in monasterio, qui non discat literas et de scripturis 
aliquid teneat.” 

*1b:, 10:(1), py 14. 


HISTORICAL REASONS FOR TEACHING ARITHMETIC 5 


For the medizeval cloister schools the computation 
of Easter day was the one great problem. On this 
depended the other movable feasts, and every monastery 
was under the necessity of having someone who knew 
enough of calculating to determine this date.! 

Utilitarian among trading peoples— Among the 
Semitic peoples we find arithmetic more extensively 
taught. The Semite has generally interested himself 
not in the thing for its own sake, but for what it 
contained for him in a practical way. Hence the 
Assyrians and Arabs and related peoples have no 
national epos and no enduring art.2 But they found 
in arithmetic a subject usable in trade, and hence it 
was extensively taught in their schools. Among the 
ruins in and about ancient Babylon it is not uncom- 
mon to find tablets containing extensive bank ac- 
counts, and lately some interesting specimens of 
pupils’ work in arithmetic have come to light.® 

Among the Jews, after elementary instruction was 
made obligatory,* arithmetic formed, with writing and 
the study of the Pentateuch, the sole work from the 
sixth to the tenth year of the child’s school life. 


1 Rashdall, H., Universities of Europe in the Middle Ages, I, p. 35. 
Schmid, II (1), p. 117. 

2 Schmid, I, p. 142. 

BAA, D152, 153. The firm of Egibi and Sons is often mentioned in 
these tablets; it was long famous in banking business from Nebuchadnez- 
zar’s time on. 

4 A.D. 64. Laurie, p. 97. 


6 THE TEACHING OF ELEMENTARY MATHEMATICS 


Even in Greece, and among the _ philosophers, 
where one would expect something beyond the mere 
necessities of existence, arithmetic was not in general 
highly valued. Socrates, who recommends the sub- 
ject in the curriculum, does so with a warning against 
carrying it beyond the needs of common life. Of 
course among the Spartans, who trained for war, the 
science had no place.! 

In Rome, a city of commerce and of war, the sub- 
ject was naturally looked upon as of merely utilita- 
rian importance. The vast commercial interests of 
the city, extending to the farthest corner of the 
great empire, made a business education imperative 
for a large class. Arithmetic flourished, but merely 
as the drudgery of calculation. So Cicero tells us 
that in his time the Romans esteemed only practical 
reckoning, nor was the learned Boethius, the philo- 
sopher, ecclesiastic, and mathematician, able to raise 
it to any higher plane.? 

In the cloisters, when not taught for the purposes 


1 Girard, Paul, L’Education Athénienne au V® et au IV® siacle avant 
J. C., 2. éd. Paris, 1891, p. 136-138; Martin, Alex., Les Doctrines 
Pédagogiques des Grecs, Paris, 1881, p. 12; Schmid, I, p. 231, 232. 

2 Laurie, p. 360; Clarke, G., The Education of Children at Rome, New 
York, 1896, p. 16, 17, 85; Sterner, M., Geschichte der Rechenkunst, 
Miinchen, 1891, p. 73, hereafter referred to as Sterner, Schmidt, K., 
Geschichte der Padagogik, Céthen, 1873, I, p. 408; Dittes, F., Geschichte 
der Erziehung und des Unterrichts, 9. Aufl., Leipzig, 1890, p. 73 ; Schmid, 
II (1), p. 140. 


HISTORICAL REASONS FOR TEACHING ARITHMETIC 7 


of computing Easter or as a “whetstone of wit,” 
arithmetic was considered as merely of value in 
trade. Even Beda, one of the best teachers of his 
time, looked upon the subject as purely utilitarian.? 
During the middle ages, too, there was a great 
revival of trade and a corresponding revival of com- 
mercial arithmetic. For a long time after the close 
of the thirteenth century Northern Italy was the 
gateway for trade entering Europe from the Orient. 
Thence it passed northward, through Augsburg, 
Nirnberg, and Frankfurt am Main, to Leipzig and 
the northern Hanseatic towns on the east, and to 
Cologne and the Netherlands on the west. Similarly 
in France, Lyons and Paris, and in Austria, Vienna, 
Linz, and Ofen, became important commercial cen- 
tres. But Italy was par excellence the mercantile 
nation and the source of commercial arithmetic, and 
we find the utilitarian influence supreme, from the 
source all along this pathway of commerce? It was 
among the merchants along this path of trade that 
as early as the thirteenth century a feeling of dis- 
satisfaction arose against the arithmetical training 
of the Church schools. Mysticism and formalism 
had so supplanted religion, to say nothing of other 


1 Schmid, II (1), p. 140. 

2 Unger, F., Die Methodik der praktischen Arithmetik in historischer 
Entwickelung vom Ausgange des Mittelalters bis auf die Gegenwart, 
Leipzig, 1888, p. 3 seq. Hereafter referred to as Unger. 


8 THE TEACHING OF ELEMENTARY MATHEMATICS 


subjects of study, that even the common people were 
wont to point with shame to the results of monastic 
training.! Even when the universities began to 
spring up, about 1100,? and arithmetic might hope to 
break away from the bonds of commerce, there was 
little improvement. Scholasticism, disputations, philo- 
sophic hair-splitting —these had little use for a sub- 
ject like this. One who had made a little progress 
in fractions was a mathematician. Save as leading 
to the calculations of the calendar, and as it might 
occasionally touch the Aristotelian Pa OSB mathe- 
matics had no standing.® 

“Tt was during this medizval period that the ‘sHan- 
“seatic league became a power. This great trust — for 


iocnmid, Le (1), Pp» 302; 
2 Laurie, S. S., The rise of Universities, lect. vi. 
8 «Omnis hic excluditur, omnis est abiectus, 
Qui non Aristotelis venit armis tectus.” } 
Chartular. Univ. Paris, I, Introd., p. xviii. 


Schmid, II (1), p. 427, 447, 448. In Cologne in 1447 the outlook for 
mathematics, as indeed for other subjects, was exceedingly poor if one 
may judge from the verses in Horatian measure of the young Conrad 


Celtes : 
“Nemo hic latinam grammaticam docet, 


Nec explotis rhetoribus studet, 

Mathesis ignota est, figuris 

Quidque sacris numeris recludit. 

Nemo hic per axem candida sidera 

Inquirit, aut quee cardinibus vagis 

Moventur, aut quid doctus alta 

Contineat Ptolemzeus arte.” — Schmid, IT (1), p. 449. 


HISTORICAL REASONS FOR TEACHING ARITHMETIC 9 


such it may be styled — soon found that it was neces- 
sary to establish its own schools if it wished a prac- 
tical education for the rising generations. And so 
there was to be found in each town of any size along 
the highway dominated by the league, an arithmetic 
master (Rechenmeister), who held the monopoly of 
teaching the subject there. Not unfrequently was 
the Rechenmeister also the city accountant, treasurer, 
sealer of weights and measures, etc. It was natural, 
therefore, that arithmetic should tend to become a 
purely utilitarian subject in these places, and so in 
great measure it was. It is interesting to recall that 
the last of the Rechenmeisters, Zacharias Schmidt of 
Niirnberg, kept his place until 1821.1 As late as the 
sixteenth century, when the reformers began to do 
some thinking in education, in a school as famous as 
the Strassburg gymnasium, Johann Sturm, in his cur- 
riculum of 1565, makes no mention of arithmetic in 
his entire ten years’ course, so completely commer- 
cial had the subject become.? 

To refer more specifically to the universities, even 
at Cambridge, which already in the middle ages led 
Oxford in mathematical teaching, arithmetic had 
scarcely any attention.* At Oxford during this period 


1 Unger, p. 26, 33. 

2 Paros, Jules, Histoire universelle de la Pédagogie, p. 126; Schmid, 
LE G2 )S p..325. 

3 Rashdall, H., Universities of Europe in the Middle Ages, II, p. 556. 


"Weg 


10 THE TEACHING OF ELEMENTARY MATHEMATICS 


a term in Boethius was all that was required.1 Even 
when a chair of arithmetic was founded in the Uni- 
versity of Bologna, a school which owed its promi- 
nence in mathematics to Arabo-Greek influence, it 
was little more than that of a surveyor and general 
computer. In Paris the subject had no hold, and in 
Vienna, where more was done than in the Sorbonne, 
only a nominal amount of arithmetic was required.? 
In general, mathematics was looked upon as a light 
subject in the medizval universities. 

Tradition and examinations— The Egyptian reason 
for teaching arithmetic may be seen in the interesting 
account of a school of the fourteenth century B.c., 
given by the late Dr. Ebers in the second chapter of 
Uarda.* Here, where the life and thought of the 
people, so closely joined to the river with its periodic 
mystery of rise and fall, naturally took on regularity, 
rule, canonical form, and mysticism, educational prog- 
ress could only come from renewed intercourse with 
the outer world. Hence arithmetic came to be taught 
merely as a matter of custom, of tradition as fixed as 
human law can be. It was required for examinations, 


and the examiner followed a certain line; hence, the 


1 Rashdall, H., Universities of Europe in the Middle Ages, II, p. 457. 

ZAb. + 1L,) Pp.) 243,°000 Desitdy Tezno: 

8 For the B. A. degree, “ Primum librum Euclidis . . . aliquem librum 
in arithmetica.” Ib., II, p. 240, 674. 

#See also Schmid, I, p. 172. 


HISTORICAL REASONS FOR TEACHING ARITHMETIC [II 


student must be prepared along that line! This is 
always the tendency under a centralized examina- 
tion system, or where an inflexible official programme 
must be followed. As M. Laisant says, “a_pro- 
gramme is always bad, essentially because it is a 
programme.” 

An excellent illustration of the petrifying tendency 
of such an examination system has recently come to 
light. The oldest deciphered work on mathematics 
is a papyrus manuscript preserved in the British 
Museum. It was copied by one Ahmes (Aahmesu, 
the Moonborn), a scribe of the Hyksos dynasty, say 
between 2000 and 1700 B.c., from an older work dat- 
ing from 2400 B.c.2~ Without going into details as 
to the contents of the work, it answers the present 

purposes to say that the arithmetical part was de- 
voted chiefly to unit fractions. Instead of writing the 
fraction =45 (using modern notation) Ahmes and his 

redecessor write it +, + + +-=1,. Now, within the 
P eG Beaty: 11f 
past decade there have been found in Kahun, near 

1 Schmid, I, p. 173; Laurie, p. 44. 

2 That is, from the reign of Amenemhat III, 2425-2383 B.c. Cantor, 
M., Vorlesungen iiber Geschichte der Mathematik, I, p. 21, n. This work, 
the standard authority in the history of mathematics, will hereafter be 
referred to as Cantor, Vol. I, 2. Auf., 1894, Vol. II, 1892, Vol. III, 1898, 
Leipzig. The Ahmes papyrus was translated and published by Eisenlohr, 
A., Ein mathematisches Handbuch der alten Aegypter, Leipzig, 1877, 
and an English edition has recently appeared. A brief summary is given 


in Gow, J., A short History of Greek Mathematics, Cambridge, 1884, p. 15, 
hereafter referred to as Gow. 


12 THE TEACHING OF ELEMENTARY MATHEMATICS 


the pyramids of Illahum, two mathematical papyri 
treating fractions exactly after the manner of Ahmes, 
and there has been published in Paris an interesting 
papyrus found in the necropolis of Akhmim, the 
ancient Panopolis, in Upper Egypt, written by a 
Christian Greek somewhere from the fifth to the ninth 
century A.D. In this latter work, also, fractions are 
treated just as Ahmes had handled them -over two 
thousand years before! The illustration is extreme, 
but it shows the tendency of tradition, of canonical 
laws, and of the examination system, which for so 
many centuries dominated the civil service of Egypt. 

The culture value — Occasionally, however, even in 
ancient times, there appeared a suggestion of a higher 
reason for the study of arithmetic. Solon and Plato 
saw in the subject an opportunity for training the 
mind to close thinking, the former placing here its 
greatest value, and the latter asserting that even the 
most elementary operations contributed to the awaken- 
ing of the soul and to stirring up “a sleepy and un- 
instructed spirit. We see from the Platonic dialogues 
how mathematical problems employed the mind and 
thoughts of young Athenians.”’? Plato even goes so 


1 Baillet, J.. Le papyrus mathématique d’Akhmim, Paris, 1892, in the 
Mémoires . . . de la mission archéologique fran¢aise au Caire. 

2 Browning, Oscar, Educational Theories, New York, 1882, p. 6; Mar- 
tin, Alexandre, Les Doctrines Pédagogiques des Grecs, Paris, 1881, p. 44; 
Schmid, I. p. 233. : 


HISTORICAL REASONS FOR TEACHING ARITHMETIC 13 


far as.to wish arithmetic taught to girls, and Aristotle 
also champions the higher cause when he asserts that 
“children are capable of understanding mathematics 
when they are not able to understand philosophy.” 
Still, in Aristotle’s scheme of state education we look 
in vain for any details as to the carrying out of the 
idea here expressed.t Naturally, too, Pythagoras, the 
first great mathematical master, saw in arithmetic 
something beyond mere calculation. “Gymnastics, 
music, mathematics, these were the three grades of his 
educational curriculum. By the first the pupil was 
strengthened; by the second purified; by the third 
perfected and made ready for the society of the 
gods.”’? 

In the middle ages the same feeling occasionally 
crops out, as when AZneas Sylvius (later Pope Pius 
II, from 1458 to 1464), the apostle of humanism in 
Germany, advocated the study of arithmetic for its 
own sake, provided it should not require too much 
time. Humanism failed, however, to advance math- 
ematics to any great extent in the learned schools. 
With few exceptions this task was left to the tech- 
nical schools. Occasionally some leader like Stehn 
was far-sighted enough to appreciate in a slight 


1 Davidson, Thomas, Aristotle and Ancient Educational Ideals, New 
York, 1892, p. 198. 

2Tb., p. 100. But see Mahaffy, P. J., Old Greek Education, New 
York, 1882, p. 89, on the slight influence of Pythagoras on education. 


14. THE TEACHING OF ELEMENTARY MATHEMATICS 


degree the educational value of the subject, but such 
cases were rare.! 

As a remunerative trade—In the development of 
the science there have been periods in which it was 
not uncommon for mere problem-solvers to undertake 
arithmetical puzzles for pay, and occasionally arith- 
metic has been studied with this in view, although of 
course to no great extent. Hans Conrad, a friend of 
Adam Riese the famous German arithmetician (1492- 
1559), solved problems for pay. Also in the time of 
the early Italian algebraists, Scipione del Ferro, An- 
tonio del Fiore, Tartaglia, and Cardan, the same state 
of affairs existed; it was a period of secret rules, and 
learning was néither open nor free.” 

As a mere show of knowledge — This has not unfre- 
quently been one of the most apparent of reasons, and 
especially so in the Latin schools of the sixteenth cen- 
tury. Thus Gemma Frisius, one of the most famous 
text-book writers of his time, presents as the second 
number in his arithmetic, 23456345678, “vicies & ter 
millies millena millia, quadringenta quinquaginta sex 
millena millia, trecenta quadraginta quinque millia, 
sexcenta & septuaginta octo.’’ 8 Such a display of words 


1 Stehn (Johannes Stenius) writes, in Wittenberg in 1594, “Num dis- 
ciplina numerorum Methodica iure possit exulare Scholis puram et solidam 
Philosophiam ambientibus.” Schmid, II (2), p. 373. 

2 Unger, p. 33, 34. 

3 Arithmeticae Practicae Methodus Facilis, edn. of 1551, p. A. v. 


HISTORICAL REASONS FOR TEACHING ARITHMETIC 15 


cannot be dignified by the term knowledge; it is only 
a pretence. It has its counterpart in the absurdly ex- 
tended number names in some of our present arith- 
metics and in subjects like compound proportion. 

As an amusement — Arithmetic has also been taught 
for its amenities, and in the seventeenth century 
several works appeared with this avowed purpose. 
Such was one published anonymously in Rouen in 
1628, ‘‘Recréations mathématiques composées de 


” 


plusieurs problemes d’Arithmetique, etc.” Schwen- 
ter’s “ Deliciz Physiko-Mathematicz oder mathema- 
tische und physikalische Erquickstunden” (Altdorf, 
1636) was another. Perhaps the best known was 
Bachet de Meéziriac’s ‘‘ Problemes plaisants et délect- 
ables,’ which appeared in 1612,! the source of several 
of the problems which still float around our lower 
schools. 

As a quickener of the wit — Closely allied to one or 
two of the reasons already mentioned is the idea that 
arithmetic is especially fitted to make one sharp, 
keen, quick-witted. This was one of the leading 
reasons in certain of the cloister schools, the subject 
being there taught for its bearing upon the training 
of the clergy in disputation. Hence arose a masg 
of catch-problems, problems intended for argument, 
problems containing some trick of language, etc. 
Such is the famous one of the widow to whom the 


1 Fifth edition, Paris, 1834, 


16 THE TEACHING OF ELEMENTARY MATHEMATICS 


dying husband left two-thirds of his property if the 
posthumous child should be a girl, and one-third if it 
should be a boy, the remainder in either case to the 
child; the widow giving birth to twins, one of each 
sex, required to divide the property. This particular 
problem appeared in a collection of about 1000 «.D., 
and is traced back even to Hadrian’s time and the 
schools of law.1 The title of Alcuin’s (735-804) book, 
‘“‘Propositiones ad acuendos iuvenes,” and of Recorde’s 
“The Whetstone of Witte” (1557) show that for the 
space of nearly a thousand years these problems which 
were largely the product of “the empty disputations 
and the vain subtleties of the schoolmen”’ had their 
strong advocates. 

In the eighteenth century, when the reasons for 
teaching the subject began to be considered more 
scientifically, this idea was brought prominently to the 
front by a number of leaders of educational thought. 
Thus Hiibsch, who certainly deserves to rank among 
these leaders, remarks that “arithmetic is like a whet- 
stone, and by its study one learns to think distinctly, 
consecutively, and carefully.” 2 

This is still thought by certain conscientious teachers 
to be the end in view in teaching arithmetic. This 
being postulated, they seek to make arithmetical 
reasoning unnecessarily obscure and difficult, allow- 
ing the use of no equation forms, however simple and 


1 Cantor, I, p. 523, 788. 2 Arithmetica portensis, 1748, 


HISTORICAL REASONS FOR TEACHING ARITHMETIC 17 


helpful. They simply conceal the equation in a mass © 
of words, and cut off the direct path for the sake of © 
the exercise derived from stumbling over a circuitous 
route. This appears in the subject of compound pro- 
portion and in certain methods of treating percentage. 
The argument upon this point of making arithmetic 
unnecessarily hard, begun in Germany over a cen- 
tury ago,! is, if we may judge by recent American 
and German text-books, coming to a_ settlement 
in two countries at least. England, more conserva- 
tive, and France, less open minded in her lower 
schools, still attempt to draw a rigid line between 
algebra and arithmetic, thus perpetuating the diff. 
culties of the latter. 

Scientific investigation of reasons— About the close 
of the eighteenth century the reasons for studying 
mathematics began to be more scientifically considered. 
The necessity for the subject in the training of all 
classes of people began to be generally recognized. 
Arithmetic now began to be looked upon as a subject 
not for the scientist and the merchant only, but for the 
soldier, the priest, the laborer, the lawyer, and generally 
for men in all walks of life, and a subject valuable in 
various ways in the mental equipment of the youth? 
It was to train for business, but not that alone; to be 


1 Unger, p. 163. 
2 The reasons as then considered are set forth by Murhard, System der 
Elemente (1798), quoted at length by Unger, p. 142 seq. 


Cc 


18 THE TEACHING OF ELEMENTARY MATHEMATICS 


interesting, but not that alone; to train the child to 
accuracy, to correlate with other subjects, to pave the 
way for science, but none of these alone. The devel- 
opment and strengthening of the mental powers in 
general, this was Pestalozzi’s broad view of the aim 
in teaching arithmetic. ‘‘So teach that at every step 
the self-activity of the pupil shall be developed,” was 
Diesterweg’s counsel.! 

Thus with the nineteenth century the self-activity 
and independence of the pupil come to the front in | 
education. The atmosphere begins to clear. Out of 
the many reasons for the study of arithmetic two for- 
mulate themselves as prominent, reasons as yet hidden 
from the mechanical teacher, who is content with an 
answer reached by some mere rule of memory and with 
the recital of a few score of ill-understood definitions or 
useless principles, but reasons which are leavening the 
mass and which will give us vastly improved work in 
the next generation. 


1 Diesterweg and Heuser’s Methodisches Handbuch fiir den Gesammt- 
unterricht im Rechnen, 3 Aufl., 1839. 


CHAPTER II 


._.Wuy ARITHMETIC IS TAUGHT AT PRESENT 


& 


Two general reasons— In Chapter I a brief survey 
of the evolution of the reasons for teaching arithmetic 
has been given. It has there appeared: that it is 
not at all settled that the subject should have the 
time now assigned it in the curriculum, or that it 
should be taught for the purpose now in view, or (as 
a consequence) that it should be taught as we now 
teach it. | 

When we come to examine the question of the real 
reason for the study of mathematics to-day, we find 
that we seek a receding and an intangible something 
which quite baffles our attempts at capture. Indeed, 
we may rather congratulate ourselves that this is the 
case, and say with one of our contemporary educators, 
“For one, I am glad we cannot express either quanti- 
tatively or qualitatively the precise educational value — 

of any study.”’} 

In a general way, however, we may summarize the 
reasons which to the world seem valuable, by saying 

1 Hill, F. A., The Educational Value of Mathematics, Educational 
Review, IX, p. 349. 

19 


20 THE TEACHING OF ELEMENTARY MATHEMATICS 


that arithmetic, like other subjects, is taught either 
(1) for its utility, or (2) for its culturet Under the 
former is included the general “ bread-and-butter value”’ 
of the subject and its applications; under the latter, 
its training in logic, its bearing upon ethical, religious, 
and philosophical thought. 

No one will deny that arithmetic is taught for these 
two reasons. It has a bread-and-butter value because 
we need it in daily life, in our purchases, in comput- 
ing our income, and in our accounts generally. It 
has a culture value because, if rightly taught, it trains 
one to think closely and logically and accurately. 

The utility of arithmetic overrated— Since the 
school requires the pupil to spend eight or nine years 
in studying arithmetic, the general impression seems 
to be that this is because arithmetic is so useful as to 
demand so great an expenditure of time. This view 
cannot, however, be justified. ‘The direct utilitarian 
value of arithmetic—its value to the breadwinner 
—has been much overestimated; or, perhaps, it 
is nearer the truth to say that, while accuracy and 

1 Fitch, Lectures on Teaching, 6th ed., 1884, chaps. x, xi; Payne’s trans. 
of Compayré’s Lectures on Pedagogy, p. 379; Reidt, F., Anleitung zum 
mathem. Unterricht, Berlin, 1886, p. 101; Fitzga, E., Die natiirliche 
Methode des Rechen-Unterrichtes, I. Theil, Wien, 1898, p. 44, hereafter 
referred to as ttzga,; Stammer, Ueber den ethischen Wert des mathemat. 
Unterrichts, in Hoffmann’s Zeitschrift, XXVIII, p. 487, and other articles 
in this journal. The best of the recent discussions is given in Knilling, 


R., Die naturgemasse Methode des Rechen-Unterrichts in der deutschen 
Volksschule, II. Teil, Miinchen, 1899. 


WHY ARITHMETIC IS TAUGHT AT PRESENT 21 


speed in simple fundamental processes have been 
underestimated, the value of presenting numerous and 
varied themes in pure arithmetic, and of pressing each 
to great and difficult lengths, has been seriously over- 
rated.” } 

For the ordinary purposes of non-technical daily 
life we need little of pure arithmetic beyond (1) count- 
ing, the knowledge of numbers and their representa- 
tion to billions (the English thousand millions), (2) 
addition and multiplication of integers, of decimal frac- 
tions with not more than three decimal places, and 
of simple common fractions, (3) subtraction of inte- 
-gers and decimal fractions, and (4) a little of division. 
Of applied arithmetic we need to know (1) a few 
tables of denominate numbers, (2) the simpler prob- 
lems in reduction of such numbers, as from pounds 
to ounces, (3) a slight amount concerning addition 
and multiplication of such numbers, (4) some simple 
numerical geometry, including the mensuration of rec- 
tangles and parallelepipeds, and (5) enough of per- 
centage to compute a commercial discount and the 
‘simple interest on a note. 

The table of troy weight, for example, forms part 
of the technical education of the goldsmith, the tables 
of apothecaries’ measures form part of the technical 
education of a drug clerk or a physician, equation of 
payments may have place in the training of a few 


1 Hill, F. A., in Educational Review, IX, p. 350. 


22 THE TEACHING OF ELEMENTARY MATHEMATICS 


bookkeepers, but for the great mass of people these 
time-consuming subjects have no_ bread-and-butter 
value. How many business men have any more 
occasion to use the knowledge of series which they 
may have gained in school, than to use the differen- 
tial calculus? - The same question may be asked con- 
cerning cube root, and even concerning square root; 
most people who have occasion to extract these roots 
(engineers and scientists) employ tables, the cumber- 
some method of the text-book having long since passed 
from their minds. A like question might be raised 
respecting alligation, only this has happily nearly dis- 
appeared from American arithmetics, although it still 
remains a favorite topic in Germany. Equation of 
payments, compound interest (as taught in school), 
compound (and even simple) proportion, greatest com- 
mon divisor, complex fractions, and various other 
chapters are open to the same inquiry. These sub- 
jects, which are the ones which consume most of the 
time in the arithmetic classes of the grades after the 
fourth, are so rarely used in business that the ordi- 
nary tradesman or professional man almost forgets 
their meaning within a few months after leaving 
school. 

Of compound numbers, which occupy a year of 
the pupil’s time in school (a year saved in most 
civilized countries except the Anglo-Saxon, by the 
use of the metric system), the amount actually needed 


WHY ARITHMETIC IS TAUGHT AT PRESENT 23 


in daily life is very slight. The common measures 
of length, of area, of volume (capacity), and of avoir- 
dupois weight are necessary. One also needs to be 
able to reduce and to add compound numbers, but 
rarely those involving more than two or three de- 
nominations. For practical purposes a problem like 
the following is useless: Divide 2 lbs. 7 oz. 19 pwt. 
by 5 0z.6 pwt. 12 gr. 

Most of the problems of common fractions are very 
uncommon. In business and in science, common frac- 
tions with denominators above 100 are rare, the decti- 
mal fraction (which has now become the “common” 
one) being generally used. 

What, then, should be expected of a child in the 
way of the utilities of arithmetic? (1) A good work- 
ing knowledge of the fundamental processes set forth 
on p. 21; (2) accuracy and reasonable rapidity, sub- 
jects which will be discussed later in this work; and 
(3) a knowledge of the ordinary problems of daily 
life. Were arithmetic taught for the utilities alone, 
all this could be accomplished in about a third of the 
time now given to the subject. 

The culture value — Although it is true that a large 
part of our so-called applied or practical arithmetic is 
not generally applicable to ordinary business, and 
hence is quite impractical, it by no means follows 
that it may not serve a valuable purpose. ‘“ Hamlet”’ 
may bring us neither food nor clothing, and yet a 


24. THE TEACHING OF ELEMENTARY MATHEMATICS 


knowledge of Shakespeare’s masterpiece is valuable 
to every one. It is a matter of no moment in the 
business affairs of most men that they know where 
the Caucasus Mountains are, or which way the Rhine 
flows, or who Cromwell was, and yet we cannot 
afford to be ignorant of these facts. 

How, then, can the teaching of arithmetic beyond 
the mere elements be justified? Fitch, in his ‘ Lectures 
on Teaching,” already cited, puts the case tersely. He 
says, “ Arithmetic, if it deserves the high place that 
it conventionally holds in our educational system, 
deserves it mainly on the ground that it is to be 
treated as a Jlogical exercise.’ Bain remarks in the 
same tenor: “All this presupposes mathematics in 
its aspect of training; or, as providing forms, meth- 
ods, and ideas, that enter into the whole mechanism 
of reasoning, wherever that takes a scientific shape. 
As culture imposed upon every one, ¢hzs zs zts highest 
justification. But, if so, these fruitful ideas should be 
made prominent in teaching ; that is, the teacher should 
be conscious of their all-penetrating influence. More- 
over, he should keep in view that nine-tenths of pupils 
derive their chief benefit from these ideas and forms 
of thinking which they can transfer to other regions 
of knowledge; for the large majority the solution of 
problems is not the highest end.” 4 


1 Bain, A., Education, p. 152. See also Fitzga, p. 27; Rein, Pickel 
and Scheller, Theorie und Praxis des Volksschulunterrichts, I, p. 350. 





WHY ARITHMETIC IS TAUGHT AT PRESENT 25 


In other words, it seems advisable to give the child 
some training in logic. But logic as a science is too 
abstract for him. Hence the school substitutes that 
subject, which, at the time, offers the best oppor- 
tunity for this training. This is the more valuable, 
in that there is incidentally accomplished another 
result, the keeping of the numerical machinery in use 
while the child is in school, so that his powers of cal- 
culating will be unimpaired from inactivity when he 
./leaves. Arithmetic is well chosen for this training 
in logic, because it furnishes almost the only example 
of an exact science below the high school, as the 
- American courses are usually arranged. And although 
induction is more valuable to the child than deduction, 
and while it must be the keynote of primary arithmetic, 
deduction plays an important part in the latter portion 
of the subject. The fact that the child finds a posi- 
tive truth, an immutable law, at the time in his develop- 
ment when he is naturally filled with doubt, with the 
desire to investigate, and with the feeling that he 
must put away childish things, has a value difficult 
properly to appreciate. He is not sure that every 
flower has petals, that every animal needs oxygen, 
that “most unkindest” is bad grammar, or that 
Columbus was the real discoverer of America; but 
he is sure, and no argument can shake his faith, that 
whatever may happen to the universe in which he 
lives, (a + 0)? will always equal a#+ 2ab + 


26 THE TEACHING OF ELEMENTARY MATHEMATICS 


So arithmetic may, even by obsolete problems, train 
the mind of the child logically to attack the every-day 
problems of life. If he has been taught to ¢hzw& in 
solving his school problems, he will think in solving 
the broader ones which he must thereafter meet. 
The same forms of logic, the same attention to detail, 
the same patience, and the same care in checking 
results exercised in solving a problem in greatest 
common divisor, may show itself years later in com- 
merce, in banking, or in one of the learned profes- 
sions. Hence, arithmetic, when taught with this in mind, 
gives to the pupil not knowledge of. facts alone, but 
that which transcends such knowledge, namely, power. 

It must not, however, be thought from its name that 
this culture phase of the subject is of value only as 
a luxury, like the ability to dabble in music or paint- 
ing. Just because it is the child of the man in poor 
or moderate circumstances who must make his own 
way in the world, it is for the common people that 
this culture phase is most valuable. 

Teachers generally fail here — The lower elementary 
teacher of arithmetic is usually more successful than 
the one in thevhigher erades..s There are {several 
reasons for this—the primary part of the subject 
has been much better investigated, better books have 
been written about it, good higher arithmetics are 
rare, and the child in the lower grades has not to 
face the nervous shock which comes a little later; 


WHY ARITHMETIC IS TAUGHT AT PRESENT 27 


but one of the chief reasons is that the primary . 


teacher knows why she is teaching arithmetic, while 
often the one in the higher grades does not. In the 
first grade the subject is being taught largely for its 
utilities, and induction plays the important part; this 
the teacher knows and hence she succeeds. In the 
seventh grade the teacher is apt to think that induc- 
tion still plays the leading rdéle, an error which gives 
rise to much poor teaching. 

Recognition of the culture value — This culture value 
is brought out first by letting the amount taken on 
authority of the book or the teacher be a minimum. 
“Tn education the process of self-development should 
be encouraged to the uttermost. Children should be 
led to make their own investigations and to draw their 
own inferences. They should be told as little as 
possible, and induced to discover as much as posst- 
ble. ... Any piece of knowledge which the pupil 
has himself acquired, any problem which he has 
himself solved, becomes by virtue of the conquest 
much more thoroughly his than it could else be.’’} 

This is not to be construed to mean that nothing 
is to be taken for granted. We must assume, for 
example, that equals result from adding equals to 
equals. But when Euclid was criticised for proving 
that one side of a triangle is less than the sum of 
the other two, as having proved what even the beasts 


1 Spencer, Education. 


/ 


28 THE TEACHING OF ELEMENTARY MATHEMATICS 


know, his disciples were entirely right in saying that 
they were not merely teaching~ facts, but were en- 
gaged in the far more important work of giving the 
power to prove the facts. As Bain puts it, referring 
to the higher grades, “The pupil should be made 
to feel that he has accepted nothing without a clear 
and demonstrative reason, to the entire exclusion of 
authority, tradition, prejudice, or self-interest.” ? 

What, then, shall be said of text-books which give 
long lists of “ Principles” as a kind of inspired reve- 
lation to pupils? So far as these are statements of 
business customs they have place; but they are gener- 
ally theorems, capable of easy proof, and of no great 
value without this proof. 

Furthermore, if we would make a clear thinker 
of the pupil, he should not be compelled to learn, 
verbatim, all or even a majority of the definitions of 
the text-book. This does not exclude those which 
are true and understandable and valuable in subse- 
quent work; but it refers to those which are false, 
unintelligible, and not usable, and to partial definitions 
in all cases where the memorizing of the same hinders 
the comprehension of the complete definition subse- 
quently. For example, what teacher of arithmetic can 
define xumber in such way as to have the definition 
both true and intelligible to young pupils, those below 
the high school? And if he could do so, of what 


1 Education, p. 149. 


WHY ARITHMETIC IS TAUGHT AT PRESENT 29 


value would it be? Or who would care to undertake 
the definition of quantity?! The fact is that the 
simpler the term the more difficult the definition. 
Since a definition must explain terms by-the use of 
terms more simple, it follows that one must sometime 
come to terms incapable of definition.2 In daily life 
we do not learn definitions verbatim; if asked to 
define Horse, the definition would probably include the 
mule and zebra and numerous others of the equine 
family. The usual definition of multiplication has 
hindered the work of many a child in fractions, and 
yet, even in the first grade he multiplies by the frac- 
tion 4. While it is true that partial truths precede 
complete ones, it is poor teaching to impress this partial 
truth on the mind so indelibly, by a memorized state- 
ment, as to make the complete truth difficult of as- 
similation. For example, a teacher drills a class to 
memorize the fiction that if the second term of a 
proportion is less than the first, the fourth must be 
less than third, —a statement entirely unnecessary in ~ 
the logical treatment of proportion, and then, when 
the pupils come to meet I: — 2 = — 2:4, they are lost. 
To test the matter a little further, let any reader 
1 Those who may be ambitious to make the attempt might first read 
Laisant, La Mathématique, Paris, 1898, p. 14, hereafter referred to as Lazsant, 
or the simple definition of number in the Encyklopadie der mathematischen 
Wissenschaften, I. Heft, Leipzig, 1898, now in process of publication. 


2 Duhamel, J.-M.-C., Des Méthodes dans les Sciences de Rai- 
sonnement. [iére partie, 3i¢me éd., Paris, 1885, p. 16. 


30 THE TEACHING OF ELEMENTARY MATHEMATICS 


repeat the definition of mzwmber, as it was once burnt 
into his memory, and see if w(=3.14159°:-) is a 
number according to this definition, — or V2, or V— 1. 
Or try the definition of avzthmetic and see if, by this 
statement, the table of avoirdupois weight is any part 
of the subject. Does the definition of multiplication, 
as usually memorized, cover even the simple case of 
2 x §, to say nothing of V2 x V3 or -V—1x V—3? 
By the common definition of factor is } a factor of }? 
By the definition of sgware root, as usually learned, 
have we any right to speak of the square root of 3, 
since 3 has not two equal factors? Are our arith- 
metics clear enough in statement so that the memoriz- 
ing of their definitions will tell a pupil whether the 
simple series 2, 2, 2, 2,-- is an arithmetical or a 
geometric progression, or neither? 

The old argument that learning definitions strengthens 
the memory and gives a good vocabulary, has too few 
advocates now to make it worth consideration. ‘The 
réle of the memory, certainly necessary in matters 
mathematical as elsewhere, should be reduced in a 
general way to very limited proportions in rational 
teaching. It is not the images, the figures, or the 
formulae which must be impressed upon the mind, so 
much as it is the power of reasoning.” } 


1 “Ce ne sont pas les images, figures ou formules, dont il faut surtout’ 
laisser l’empreinte dans le cerveau; c’est la faculté du raisonnement.” 
Laisant, p. I9I. 


WHY ARITHMETIC IS TAUGHT AT PRESENT 31 


This opposition, on the part of leaders in education, 
to the burdening of children’s memories, is not new. 
Locke voiced the same sentiment: “ And here give me 
leave to take notice of one thing I think a fault in the 
ordinary method of education; and that is, the charging 
of children’s memories, upon all occasions, with rules 
and precepts, which they often do not understand, and 
constantly as soon forget as given.”! ‘Teachers at 
one time believed that the first object of primary 
instruction is to cultivate the verbal memory of their 
pupils, when, in fact, the verbal memory is one of the 
few faculties of our nature which need no cultivation.” ? 
Of the two, to learn all of the definitions of a text-book 
or none, the latter plan is unquestionably the better. 

But while memorized definitions may not unfrequently 
be justified, this is rarely true of the memorized rule. 
The glib recitation of rules for long division, greatest 
common divisor, etc., which one hears in some schools 
—what is all this but a pretence of knowledge? “If 
learning is a process of gaining knowledge, that is, 
a true apprehension of realities, it excludes verbal mem- 
orizing, cramming, and everything that resolves itself 
on close scrutiny into a pretence of knowledge getting.” ® 

But not only is this old-fashioned rule-learning (un- 
happily not yet extinct) a sham; it is wholly unscientific. 
Tillich, one of the best teachers of arithmetic of the 


1 On Education, Daniel’s edn., p. 126. 2 Tate. 
8 Dr. James Sully, in the Educational Times, December, 1890. 


32 THE TEACHING OF ELEMENTARY MATHEMATICS 


first half of the nineteenth century, saw the danger of 
dogmatic rules. “It is,” he said, “just as unpsycho- 
logical to begin the teaching of arithmetic by a mass 
of inherited rules as it is senseless to try to teach lan- 
guage to children by means of mere rules of speech. 
. .. Since these rules were not independently worked 
out by the child, but are simply the memorized results 
of others’ work, it cannot but be true that the arith- 
metic of most of the pupils is a mere mechanism, and 
a distasteful one at that.” 1 So, too, Jean Macé, in his 
well-known “Arithmetic of a Grand-Papa,” remarks 
that to have a child begin with the abstract rule, follow- 
ing this by the solution of a lot of problems, is to com- 
pletely reverse the order of human development.? 
There are, however, a few rules of operation which 
must be learned for the sake of facility and speed in 
numerical calculation. Such is the rule for substituting 
another and a simpler operation for that of dividing one 
fraction by another. But this does not mean that such 
a rule is to be given as a kind of inspired dogma. It 
is quite as easy, and far more valuable, to lead the 
child to discover it for himself. Even as. far back as 
Roger Ascham this was realized, though seldom prac- 
tised. ‘We do not contemne rewles,” said he, “but 


1 Lehrbuch der Arithmetik, p. xi. In a similar line, Reidt, Fr., An- 
leitung zum mathematischen Unterricht an héheren Schulen, Berlin, 1886, 
p. 103. 

2 L’Arithmétique du Grand-Papa, giéme éd., p. 12. 


WHY ARITHMETIC IS TAUGHT AT PRESENT 33 


we gladly teach rewles; and teach them more plainlie, 
sensiblie, and orderlie than they be commonlie taught 
in common scholes.”’! And the best of summaries of 
method that has recently appeared asserts: “‘ Whoever 
would bring his pupils to intelligent computation (zu 
einen verstandnisvollen Rechnen) should develop no 
rule, but should wait until the children themselves dis- 
cover it (bis die Kinder selbst darauf kommen).” 2 

Aside from the fact that we make almost no use 
of the rules of operation in our daily computations, 
needing but a few rules of business and theorems of 
mensuration, there is the further consideration that 
the child does not like to solve by rule. To use his 
common sense is to become a discoverer, and the 
zeal for discovery is one of the inborn traits of the 
human mind. If all mathematical problems were 
solved, or if we had rules for solving them, all inter- 
est in the subject would vanish. 

Of course the same objection which exists as to 
rules exists in even greater measure as to undemon- 
strated formulae, which are merely rules put in un- 
familiar language. To fill the child’s mind with a 
list of formulae for percentage, for example, is to 
take a human soul and try to make a machine of 
it. “If one learns only by memory, and does not 
think, all remains dark.” ? 

What, then, shall be said of the educative value 


1 The Scholemaster, 2 Fitzga, p. 48. 8 Confucius. 
D 


34 THE TEACHING OF ELEMENTARY MATHEMATICS 


of the old-fashioned arithmetic which put its prob- 
lems in “cases,” each preceded by the rule? Surely 
a more mechanical device could hardly be invented. 
And yet these books exist to-day in thousands of 
schools in England and America. And if it be said 
that these books in the schools of fifty years back 
produced good arithmeticians, let it not be forgotten 
that far more time was then given to the subject. 
Good arithmeticians were produced in spite of, not 
because of, such books. 

What chapters bring out the culture value— It is not 
so much the particular chapter as the way it is taught 
that brings out the educational value of arithmetic. 
A person may have exercise in logic by studying alli- 
gation — merely indeterminate equations in an awk- 
ward medizeval form. But the best results will naturally 
come from those parts that appeal to the child’s life 
and interests. 

For example, longitude and time, a subject with 
but slight utilitarian value to most people, may be 
so taught as to have high culture value. The inter- 
est attaching to the “date line”’ and to the recent 
world-movement of “standard time,” renders the sub- 
ject a delightful one to children of a certain age. 
But its value is lost when a book gives the form 
‘(75° + 15 =5 hrs.,” since it destroys the child’s pre- 
conceived and correct ideas of the nature of division ; 
accuracy of statement and of thought have been 


WHY ARITHMETIC IS TAUGHT AT PRESENT 35 


sacrificed for a mere answer, an arithmetical birth- 
right sold for a mess of pottage. 

Similarly, “true discount” may be made interest- 
ing, and the reasoning may give rise to logical power. 
But this, like other subjects that at once occur to 
the teacher, is open to the fatal objection that it 
gives a wrong idea of business. However much the 
pupil may be warned, the name “¢rue discount” will 
cling to him, and he must learn, after his school 
days have gone by, that the true is really the false 
discount in the life he is to live. 

What may well be omitted— In considering what 
may profitably be omitted from the arithmetic of to- 
day, there is, of course, the bugbear of the examina- 
tion to be taken into account as a practical question. 
But looking at the subject from the standpoint of 
the educator rather than the coach, we have to con- 
sider what there is that appeals neither to the 
utilitarian nor to the culture value, or that is found 
wanting for other reasons. 

1. The following may be said to have little or no 
utilitarian value for the general citizen, and because 
they give a false notion of business they may also 
be rejected as undesirable exercises in logic. 

(2) Equation of payments. 

(6) Alligation (now rapidly disappearing from Eng- 
lish and American text-books, although still found in 
the German). 


36 THE TEACHING OF ELEMENTARY MATHEMATICS 


(c) Insurance, in the form usually presented in text- 
books. 

(dz) “Profit and Loss,” the text-book expression not 
having the American business meaning, and the 
problems being merely ordinary ones of simple per- 
centage, not worthy of a special chapter. 

(¢) Exchange as usually taught. If the modern 
business problems are given, with the modern ma- 
chinery for exchange, the subject is valuable. Of 
course arbitrated exchange has no value fer se for 
the ordinary citizen; it is part of the technical train- 
ing of a few brokers. 

(7) Commission and brokerage so far as the sub- 
ject relates to problems like the following: “A sends 
B $1000 with which to buy wheat on a 24% com- 
inission: how much can B invest?” 

(g) Stocks, where the problems require, as in 
many text-books, fractional numbers of shares, like 
the buying of 8% shares, or where they call for un- 
used quotations like 10942. 

(Z) Partial payments beyond the common methods 
in the state in which the pupil lives. 

(z) Annual interest, beyond the mere elements. 

(7) Compound interest, beyond the ability to find 
such interest. The banker, of course, employs tables 
whenever he has occasion to use the subject. 

(£) Compound proportion, a subject in which 
hardly a text-book problem can be found that has 


WHY ARITHMETIC IS TAUGHT AT PRESENT 37 


any practical value, in spite of the pretensions of the 
subject. As for mathematical explanation, it would be 
difficult to find a text-book which makes any attempt 
in that direction. 

(2) Problems in denominate numbers involving 
more than three denominations at a time, and those 
involving tables not needed in daily life —troy, 
apothecaries’, etc. Similarly the semi-obsolete meas- 
ures, the stone (in America), the barleycorn, the 
tun, the pipe, etc., and the technical measures, the 
square (in shingling), the perch, the quintal, etc., 
have no place in the common schools. There is, 
indeed, a somewhat serio-comic aspect of the matter 
as set forth in the Football Kreld: “A gallon isn’t a 
gallon. It’s a wine gallon, or one of three different 
sorts of ale gallon, or a corn gallon, or a gallon of 
oil; and a gallon of oil means seven and a half 
pounds for train oil, and eight pounds for some 
other oils. If you buy a pipe of wine, how much 
do you get? Ninety-three gallons if the wine 
be Marsala, ninety-two if Madeira, a hundred and 
seventeen if Bucellas, a hundred and three if Port, a 
hundred if Teneriffe. What is a stone? Fourteen 
pounds of a living man, eight of a slaughtered bul- 
lock, sixteen of cheese, five of glass, thirty-two of 
hemp, sixteen and three-quarters of flax at Belfast, 
four and twenty of flax at Downpatrick. It is four- 
teen pounds of wool as sold by the growers, fifteen 


38 THE TEACHING OF. ELEMENTARY MATHEMATICS 


pounds of wool as sold by the wool-staplers to each 
other... . Our very sailors do not mean the same 
thing when they talk of fathoms. On board a man- 
of-war it means six feet, on board a merchantman five 
and a half feet, on board a fishing vessel five feet.” ? 

Of course we may say that in America “we have 
changed all that,” and that we have no such non- 
sense. And yet many a school to-day teaches the 
children the length of the cubit, which nobody knows 
or can know, because it varied, and our various states 
have different laws and customs as to what consti- 
tutes a bushel of grain, a perch of stone, etc., and 
we are quite as unsettled with respect to many meas- 
ures as is Great Britain. 

“Of late years, there has been some reform in this 
particular (the applications of arithmetic), and a few 
of the monstrosities of the old. curriculum, notably 
our ancient enemy, duodecimals, have been thrown 
overboard. But there still remain many things, as 
taught in our schools, which occupy time that could 
better be devoted to the study of other subjects, or 
at least to a greater degree of practice in simple 
operations. ... Compound interest, compound pro- 
portion, compound partnership, cube root and _ its 
applications, equation of payments, exchange, ‘similar 
surfaces,’ and the mensuration of the trapezoid and 
trapezium, of the prism, pyramid, cone, and sphere, 


1 Educational Times, October, 1892. 


WHY ARITHMETIC IS TAUGHT AT PRESENT 39 


are proposed to be dropped from the course in the 
(Boston) grammar school.” } 

2. The following may be said to have some, and 
might have much, culture value, but should be 
omitted on other grounds.? 

(a) Series, because the subject can better be 
treated where it belongs, in algebra. 

(0) The long form of greatest common divisor 
before about the eighth grade, because it is taught 
only for its logic, and this logic is too much for the 
average child below that grade. 

(c) Compound proportion, already mentioned, be- 
cause almost no arithmetic pretends to treat it other- 
wise than by rule, and an explanation is too difficult 
for pupils—as apparently for authors. Indeed, it is 
doubtful if the child derives. much good even from 
simple proportion as usually presented. 

Relative value of culture and utility— Since it 
appears that arithmetic is taught for these two 
general reasons, a question arises as to their relative 
importance. But this it is impossible to answer. We 
lack a unit of measure. Laisant remarks® that it is 


1 Walker, F. A., Arithmetic in Primary and Grammar Schools, Boston, 
1887, p. 12. 

2 «The charge I make against the existing course of study is, that it is 
largely made up of exercises which are not exercises in arithmetic at all, or 
principally, but are exercises in logic; and, secondly, that, as exercises in 
logic, they are either useless or mischievous.” Walker, Ib., 17. 

5 La Mathématique, p. Io. 


40 THE TEACHING OF ELEMENTARY MATHEMATICS 


like asking which is the more important, eating or 
sleeping; the loss of either is fatal. The teacher 
who recognizes in the subject only its applications to 
trade, would better give up teaching; the one who 
sees in it only an exercise in logic will also fail; 
but the greatest failure comes from seeing in the 
subject neither utility nor logic, as is the case with 
the teacher who blindly follows the old-style, tradi- 
tional text-book. 

But what shall be said for the teacher who fears 
to omit certain problems which are not utilitarian and 
whose culture value is counterbalanced by the fact 
that they give a false notion of business, or to omit 
those traditional puzzles which depend for their diffi- 
culty upon their ambiguity of statement? Many a 
teacher, especially in our country schools, will confess 
to such a fear of omitting problems, lest he be ac- 
cused of inability to solve them. It would be well 
for all teachers to assist in creating a sentiment in 
favor of omitting the unquestionably superfluous or 
dangerous, and thus to avoid this weak criticism. 
It should also be understood by timid teachers that 
it is no disgrace to be unable to solve every puzzle 
that may be sent in, or even every legitimate problem. 
And for those who may feel inclined to boast that they 
have never seen a problem in arithmetic which they 
could not solve, it may be interesting and instructive to 
attempt to prove the following simple statements: 


WHY ARITHMETIC IS TAUGHT AT PRESENT 4I 


The sum of the same powers (above the second) 
of two integers cannot equal a perfect power of the 
same degree. (In the case of the second degree 
there are any number of examples, as 37+ 47= 52.) 
Fermat’s theorem. 

Every even number is the sum of two prime num- 
bers. Goldbach’s theorem. 

The consecutive integers 8 and 9 are exact 
powers; are there any other consecutive integers 
which are exact powers? Catalan. 


CHAPTER III 


How ARITHMETIC HAS DEVELOPED 


Reasons for studying the subject— The historical de- 
velopment of the reasons for teaching arithmetic has 
already been considered. For the well-informed 
teacher there remain two other historical questions 
of importance. The first relates to the development 
of the subject itself, and the second to the methods of 
teaching it. 

There are good and sufficient reasons for consider- 
ing briefly the history of arithmetic. In the first 
place, the child learns somewhat as the world learns.1 
“The individual should grow his own mathematics, 
just as the race has had to do. But I do not propose 
that he should grow it as if the race had not grown 
it too. When, however, we set before him math- 
ematics, — be it high or low, —in its latest, and most 
generalized, and most compacted form, we are trying 
to manufacture a mathematician, not to grow one.” 2 
This does not mean that the child must go through 

1 Cette longue éducation de ’humanité, dont le point de départ est si 
loin de nous, elle recommence en chaque petit enfant.— Jean Macé, 


L’Arithmétique du Grand-Papa, 4i¢me éd., p. II. 
2 Jas. Ward in the Educational Review, Vol. I, p. 100. 


42 


HOW ARITHMETIC HAS DEVELOPED 43 


all of the stages of mathematical history —an extreme 
of the “culture-epoch” theory; but what has both- 
ered the world usually bothers the child, and the 
way in which the world has overcome its difficulties 
is suggestive of the way in which the child may over- 
come similar ones in his own development. 

In the second place, the history of the subject gives 
us a point of view from which we can see with 
clearer vision the relative importance of the various 
subjects, what is obsolete in the science, and what the 
future is likely to demand. Sterner! has compared 
the teacher of to-day to a traveller who by much toil 
has reached an eminence and stops to take breath be- 
fore attempting further heights; he looks over the road 
by which he has journeyed and sees how he might 
have done better here, and made a short cut there, 
and saved himself much waste of time and energy 
yonder. So one who considers the historical develop- 
ment of arithmetic and its teaching will see how 
enormous has been the waste of time and energy, 
how useless has been much of the journey, and how 
certain chapters have crept in when they were impor- 
tant and remained long after they became relatively 
useless. He will see the subject as from a mountain 
instead of from the slough of despond which the text- 
book often presents, and he will be able, as a result, 
to teach with clearer vision, to emphasize the impor- 


1 Geschichte der Rechenkunst. 


44 THE TEACHING OF ELEMENTARY MATHEMATICS 


tant and to minimize or exclude the obsolete, and thus 
to save the strength of himself and of his pupils. 
He will also learn that some of the most valuable 
parts of arithmetic knocked at the doors of the schools 
long centuries before they were admitted, and that 
teachers have had to struggle long and persistently 
to banish some of the most objectionable matter. As 
a result, while he may condemn the conservatism 
which excludes the metric system and logarithms and 
certain of the more rational methods of operations to- 
day, he will have more faith in the ultimate success 
of a good cause and will see more clearly his duty 
as to its advocacy. 

Extent of the subject—It is manifestly impossible 
to give more than a glimpse at the history of arith- 
metic. The simple question of numeration, discussed 
with any fulness, would fill a volume the size of this 
one.! DeMorgan’s masterly little work, “ Arithmetical 
Books,” hardly more than a catalogue (with critical 
notes) of certain important arithmetics in his library, 
fills one hundred twenty-four pages.* For the stu- 
dent who cares to enter this fascinating field some sug- 
gestions are given in a subsequent chapter. But for 
the present purpose it suffices to consider merely a 
few important events in the general development of 
the subject. 


1 See, for example, Conant, L. L., The Number Concept, New York, 1896, 
2 London, 1847. 


HOW ARITHMETIC HAS DEVELOPED 45 


The first step — counting — The first step in the his- 
torical development of arithmetic was to count like 
things, or things supposed to be alike; in the broad 
sense of the term this is a form of measurement.! 
Arithmetic started when it ceased to be a question of 
this group of savage warriors being more than that, 
and began to be recognized’ that this group was three 
and that two; when it was no longer a matter of a 
stone axe being worth a handful of arrow heads, but 
one of an exchange of one axe for eight arrows. 
How far back in human history this operation goes 
it is impossible to say, just as it is impossible to say 
how far back human history itself goes. Indeed, 
counting is not limited to the human family, for 
ducks count their young and crows count their ene- 
mies.2, Any discussion of the nature of this animal 
counting must lead to the broader question of the 
ability to think without words, a matter so foreign to 
the present subject as to have no place here.? 

The race has not, however, always counted as at 
present. It was a long struggle to know numbers up 


1 In this connection the teacher should read, though he may not fully 
indorse, Chap. III of McLellan and Dewey’s Psychology of Number, 
New York, 1895. 

2 This subject of animal counting has often been discussed. It is 
briefly treated in the chapter on Counting in Tylor’s Primitive Culture, 
and also in Conant’s Number Concept mentioned on p. 44. 

8 For Max Miiller’s side of the case see his lecture on the Simplicity of 
Thought. 


46 THE TEACHING OF ELEMENTARY MATHEMATICS 


to ten. The primitive savage counted on some low 
scale, as that of two or three. To him numbers were 
“T, 2, many,” or “I, 2, 3, many,” just as the child 
often says, “I, 2, 3, 4, a lot,’ and somewhat as we 
count up very far and then talk of “infinity.” 

It is evident that there must be some systematic 
arrangement of numbers in order that the mind may 
hold the names. For example, if we had unrelated 
names for even the first hundred numbers, it would 
be a very difficult matter to teach merely their se- 
quence, to say nothing of the combinations. But by 
counting to ten, and then (or after twelve) combining 
the smaller numbers with ten, as in three-ten (thir- 
teen), four-ten (fourteen), . . . twice-ten (twenty), and 
so on, the number system and the combinations are 
not difficult. 

We might take any other number than ten for the 
base (radix). If we took three we should count, 


one, two, three, three-and-one, 
three-and-two, two-threes,..., 
and (with our present numerals) write these, 
I, 2, 3, 11 (2.e., oné three and one unit), 12; 20,< ... + 


But most peoples, as soon as they were far enough 
advanced to form number systems, recognized the 


1 A brief but interesting summary of this subject is given in Fahrmann, 
K. E., Das rhythmische Zahlen, Plauen i. V, 1896, p. 21. It is also 
treated in numerous text-books and elementary manuals in English. 


HOW ARITHMETIC HAS DEVELOPED 47 


natural calculating machine, their fingers, and hence 
began to count on the scale of ten (our decimal 
system). “In the book of Pvroblemata, attributed to 
Aristotle, the following question is asked (XV, 3): 
‘Why do all men, both barbarians and Hellenes, count 
up to 10, and not to some other number?’ It is 
suggested, among several answers of great absurdity, 
that the true reason may be that all men have ten 
fingers: ‘using these, then, as symbols of their proper 
number (viz., 10), they count everything else by this 
Slee 

To-day it is common to hear teachers object to 
allowing a child to count on his fingers. And yet 
one of our best teachers of arithmetic has just re- 
marked, what is indorsed both by history and by com- 
mon sense, that the fingers are the most natural and 
most available material? It is true that there is some 
ground for the objection, especially on the part of 
teachers who have not the ability to lead children 
to rapid oral work; but if the world had not counted 
in this way we should not have had our decimal 
system. 

It is really a little unfortunate, arithmetically con- 
sidered, that man has ten instead of twelve fingers, 


1 Gow, J., History of Greek Mathematics, Cambridge, 1884, Chap. I. 

2 Die Finger sind also das natiirlichste und nachste Versinnlichungs- 
mittel. Fitzga, p. 82, 14, 59. See also Conant’s Number Concept, p. 10, _ 
et pass. 


48 THE TEACHING OF ELEMENTARY MATHEMATICS 


for the scale of twelve is the easiest of all the scales. 
A radix must not be too small, since that would 
require too much labor in writing comparatively small 
numbers. For example, on the scale of 3, fourteen 
would appear as 112 (1-37+ 1-3 +2). Neither should 
the radix be too large, since there must be ten figures 
for the radix ten, twenty for the radix twenty, and 
so on, and too many characters are objectionable. 
Twelve, like ten, is a medium radix; but it is better 
than ten because it has more divisors. Consider, for 
instance, the fractions most commonly used, viz., 4, 


3, $+, $ These are written 


On the Stale Ol O, WP O15 nO. 33 Se Se Ol? bm da 


on the iscalevot 12, 0.6,. 0.4, 0.357) (Onl Go: 


Hence the advantage of the duodecimal scale, in all 
work involving fractions, is apparent. 

Counting must have preceded notation by many 
generations, just as talking preceded writing. And 
while there are good reasons for teaching the num- 
erals to a child while he is learning number (the 


”) 


character ‘‘3’’ while he is learning to pick out three 
things), Pestalozzi had the argument of race develop- 
ment on his side when he advocated teaching the 
characters only after the child could count to ten. 
And in teaching the child number, while it would 
be very logical to introduce the ratio idea first, —the 


idea which Newton crystallized in his well-known 


HOW ARITHMETIC HAS DEVELOPED 49 


definition of number,—-the plan is not in harmony 
with the historical development of the race; first, 
counting ; second, simple operations; third, a notation; 
this is the race order. Aside from all this, there is 
the more serious question, discussed in a subsequent 
chapter, as to the psychological phase of the matter; 
whether the ratio idea is not altogether too abstract 
for the mind of the child beginning to study num- 
ber. It can be taught, but its success means a good 
teacher with a poor method, a David with a sling. 
While the introduction of the idea in the beginning 
is unwarranted by considerations historical, and seems 
to be so by considerations psychological, it is desir- 
able as soon as the child has developed sufficiently to 
allow it. The matter has not yet been carefully 
enough investigated, however, to tell just when this 
is. Laisant, who does not lose his head in such 
affairs, questions whether the ratio idea, usually rele- 
gated to the later years of the elementary course, 
should not enter very early, but after careful con- 
sideration is forced to the conclusion that ‘“ number, 
in its elementary form, comes to us by the evaluation 
of collections of like objects.” 

The second step—notation— Of course there de- 
veloped in connection with counting a certain amount 
of calculating —the simplest operations. But the 
second step of great importance was that of writing 


1 La Mathématique, p. 30, 31. 


50 THE TEACHING OF ELEMENTARY MATHEMATICS 


numbers. The plans with which we are familiar, the 
Hindu (“Arabic”) and the Roman, are only two of 
many which have been used. The primitive one was 
that of simple notches in a stick or scratches on a 
stone. But of scientific systems there are only a few 
types. 

The Egyptians had a system much like the Roman 
in general plan, —symbols for 1, 10, 100 and higher 
powers of 10.! 

The Babylonians, not having the abundance of 
stone possessed by the Egyptians, resorted to writ- 
ing on soft bricks, which were then baked. They 
therefore developed a system which required but a 
few characters such as could easily be impressed by 
a stick upon clay, the so-called cuneiform numerals. 
Their symbols were three, —one for 1, one for 10, and 
one for 100.” | 

The early Greeks used the initial letters of the 
words for 5, 10, I00, 1000, 10,000, a plan leading to 
a system about like the Egyptian and Roman. The 
late Greeks and the Hebrews used their alphabets, 
giving to each letter a number value. Thus the 
Greeks used a for 1, 8 for 2, y for 3, 6 for 4, e for 
5, an old form called digamma for 6,-¢ for 7, » for 


1 Cantor is, of course, the standard authority on all such matters, A 
good summary is given in Sterner, p. 17 seq. 

2 They are given in Beman and Smith’s translation of Fink’s History 
of Mathematics, Chicago, 1900. 


f 


HOW ARITHMETIC HAS DEVELOPED 51 


8, and @ for 9. The next nine letters, with one 
extra symbol, stood for tens, = 10, k= 20, X= 30, 
etc., and the rest, with one extra character, for the 
hundreds. The system was a difficult one to master, 
but it enabled the computer to write numbers below 
1000 with few characters. For example, 387, which 
the Romans wrote CCCLXXXVII, the Greeks wrote 
T7104 

The Romans used a system the essential features 
of which are known to all. The origin of the symbols 
has long been a matter of dispute, but they are now 
generally recognized to be modified forms of old Greek 
letters, not found in the Latin alphabet, which came 
through the Chalcidian characters? The Romans in- 
troduced the ‘‘subtractive principle” of writing IV 
for 5—1, XL for 50—10, etc., but they and their 
successors made little use of it. The tendency to 
write IIII for IV is still seen on our clock faces. The 
bar over a number was rarely used, the number usually 
being written out in words if above thousands, while the 
double bar sometimes seen in American examination 
questions, and the idea that a period must follow a 
Roman numeral, may be called stupid excrescences of 
the nineteenth century. The fact that the Romans 


1 For more complete discussion see Cantor, I, p. 117, or Sterner, p. 50. 

2 Wordsworth, Fragments and Specimens of the Early Latin, p. 8; 
Fink’s History of Mathematics, English, by Beman and Smith, p. 12; 
Cantor, I, p. 486; Sterner, p. 78. 


52 THE TEACHING OF ELEMENTARY MATHEMATICS 


did not make practical use of their system in writing 
large numbers should show us the criminal waste of 
time in requiring children of our day to bother with 
the system beyond thousands. 

The Hindu (or so-called Arabic) system can be traced 
back to certain inscriptions found at Nana Ghat, in 
the Bombay Presidency (India), and first made known 
to the western world in 1877. These inscriptions 
probably date from the early part of the third cen- 
tury B.c.! and seem to prove that the numerals from 
4 to 9 inclusive were the initial letters of words in 
the ancient Bactrian alphabet.2 The system was at 
that time, and for several centuries: thereafter, no 
better than many others of antiquity, because it had 
no zero, without which one element of superiority, 
the place-value element, is wanting. Without the 
zero we cannot write ten, one hundred six, and so 
on. And while the place value was somewhat ap- . 
preciated as early as the time of the cuneiform nu- 
merals, the zero does not seem to have appeared in 
the Hindu system before 300 a.p.,2 and the first 
known use of the symbol in a document dates from 
four centuries later, 738 a.p.! 

There is much question as to the way in which 
the Hindu numerals first entered the western world. 


1 See Journal of the Royal Asiatic Society, 1882, N.S. XIV, p. 336; 
1884, N.S. XVI, p. 325 seq., especially 347. 
2 Cantor, I, p. 564. 8 Tb., p. 567. 4 -Ib., p..863, 


HOW ARITHMETIC HAS DEVELOPED 53 


Sporadic use of the characters is found before the 
thirteenth century. But about 1200 a.p., Leonardo 
Fibonacci, of Pisa, returning from a voyage about the 
Mediterranean, brought them to Italy. Being then 
in use in various Moorish towns, they received the 
name “ Arabic,” although the Arabs may have done 
nothing more than to disseminate them along the 
borders of the Occident. If, as is not probable,} 
they invented the zero, they deserve to have the name 
‘‘ Arabic’”’ continued, but if not, the title “ Hindu nu- 
merals’’ is much to be preferred. 

It was nearly a century later than Leonardo’s time 
before the system had penetrated as far north as 
Paris,2 and it was not until about 1500 that, thanks 
to the invention of printing, it began to get a firm 
footing in the schools.? For teachers who await with 
impatience the popular use of the metric system, or 
who are discouraged by the apathy of their co-workers 


1 Cantor, I, p. 569, 576. 

2 Henry, Ch., Les deux plus anciens Traités Francais d’Algorisme et 
de Géométrie, Boncompagni’s Bulletino, February, 1882. The Ms. is 
anonymous and was written about 1275 A.D. 

8 Those who are interested in this period of struggle, from 1200 to 1500, 
will find, besides the discussions in Cantor, Unger, Sterner, and other 
writers on history, some interesting facsimiles in Kénnecke, G., Bilder- 
atlas zur Geschichte der deutschen Nationallitteratur, Marburg, 1887, 
p- 40, et pass. Halliwell, J. O., Rara Mathematica, London, 2d. ed., 1841, 
is likewise interesting and valuable, as is also the pamphlet edition of 
“The Crafts of Nombrynge,” published in 1894 by The Early English 
Text Society. Boncompagni’s Bulletino is, of course, rich in material. 


54. THE TEACHING OF ELEMENTARY MATHEMATICS 


with respect to the use of logarithms in physical com- 
putations, the story of the struggles of the Hindu 
system is of value. 

The awkwardness of the old Roman system, in 
general use even after the opening of the sixteenth 
century, is well seen in Kobel’s arithmetic,! a work 
which barely mentions the Hindu numerals. The 


following is a specimen: “If you would add a to 


vit write them crosswise on the abacus; then by 
multiplying, III times III is IX, and II times IV 
is VIII; add the VIII and IX getting XVII, and 
this is the numerator; then multiply the denomina- 
tors, III times IIII is XII; write the XII under 
the XVII and make a little line between, thus 
XVII AV 
XII’ XIT 
as 1658, when Comenius published in Niirnberg the 


which equals one and Even as late 


first picture book for the instruction of children, the 
well-known Orbis Pictus, the Roman numerals were 
in common use, for he says, ‘The peasants count 
by crosses and half crosses (X and V).” 

The next great step in arithmetic, after the writing 
of integers, was that leading to a knowledge of frac- 
tions. The recognition of simple fractions is pre- 
historic; but the struggle to compute with fractions 
extended for thousands of years after Ahmes copied 


1 Das new Rechépiichlein, 1518, quoted here from Unger, p. 16. 


HOW ARITHMETIC HAS DEVELOPED 55 


his famous papyrus. It has already been stated 
(p. 11) that the ancient Egyptians could, in general, 
write only such fractions as had a numerator I, and 
the same is true of other ancient peoples. The later 
Greeks wrote the numerator followed by the denomi- 
nator duplicated, and all accented, thus, +6’ xa!’ xa", 
for 47.1. The Romans had a fancy for fractions with 
a constant denominator as a power of 12, as seen 
in our inch (44 of a foot), and the Babylonians for 
fractions with a denominator 60 or 60%, as seen in our 
minute and second (1/ = 7 of a degree, 1" = (4) of 
a degree). 

With such a struggle to write fractions, it is not 
to be wondered at that the ancients did relatively 
little in arithmetical computation, or that the child 
of to-day has to struggle to master the subject. The 
world could solve the simple equation many centuries 
before it could do much with fractions, and hence it is 
entirely in harmony with the world growth to introduce 
in the first grade such simple equations as 2+(?)=7 
before any work in fractions is attempted. 

The decimal fraction is a very late product of 
arithmetical ingenuity. It appeared in the sixteenth 
century, in forms like 5548 and 5 @ 7 @ 8 @, for 
0.578, and about 1592 a curve was used by Biirgi 
to cut off the decimal part. But in 1612, Pitiscus 
actually used the decimal point, and the system was 


1 Cantor, I, p. 118. 


56 THE TEACHING OF ELEMENTARY MATHEMATICS 


perfected! It was not, however, until well into the 
eighteenth century that decimal fractions found much 
footing in the schools, nor was it until the nineteenth 
century that their use became general. During the 
long struggle for supremacy, the old-style fraction 
was literally the ‘‘common fraction”; the name still 
survives, although the decimal form is now by far 
the more common. 

In educational circles we often hear advocated the 
plan of teaching decimal fractions before common 
fractions. But to attempt any theory of decimal frac- 
tions first, or to exclude the simplest common fractions 
from the first year of arithmetic, is unscientific from 
both the psychological and the historical standpoints. - 
The historical order is, (1) the unit fraction, (2) the 
common fraction (of course not in its complete de- 
velopment), and (3) the decimal fraction, and this is 
also the natural sequence from simple to complex, 
from concrete to abstract. | 

The twofold nature of ancient arithmetic— As has 
been said, arithmetic was studied by the ancients both 
as a utilitarian and a culture subject. The Greeks, 
for example, differentiated the science into Arithmetic 
(apiOuntixn) and Logistic (AoyortKy), the former hav- 
ing to do with the theory of numbers, and the latter 
with the art of calculating.*? Hence when, long after, 


1 Cantor, II, p. 566-568. 
2 Gow, J., History of Greek Mathematics, p. 22. 


HOW ARITHMETIC HAS DEVELOPED 57 


these two branches came together to form our modern 
arithmetic, the subject came to be defined as “the 
science of numbers and the art of computation,” al- 
though the modern arithmetic of the schools includes 
much besides this. 

The apiOuntixyn of the Greeks ran also into the 
mystery of numbers, and much was made of this sub- 
ject by Pythagoras (b. about 580 B.c.) and his fol- 
lowers. That “there is luck in odd numbers”’ probably 
dates back to his school, the Latin aphorism, 


“Deus imparibus numeris gaudet,” 
being much older than Virgil’s line, 
“Numero deus impare gaudet.” (Eclogue viii, 77.) 


The mysticism of numbers, the universal recognition 
of 3, 7, and 9, as especially significant, forms even 
now an interesting study. It is to this ancient ten- 
dency that we owe the study, only recently banished 
from our schools, of numbers classified as amicable, 
deficient, perfect, redundant, etc. 

The art of calculating (Aoytor“KH) among the ancients 
ran largely to the use of mechanical devices, such as 
counters (like our checkers), and the abacus, an in- 
strument with pebbles (ca/culz, whence our word calcu- 
late) sliding in grooves or on wires. To-day the 
Chinese laundryman in America still performs his 
calculations on an abacus (his swan pan), and in 
Korea the school-boy still carries to school his bag of 


58 THE TEACHING OF ELEMENTARY MATHEMATICS 


counters (in this case short pieces of bone). Among 
the ancients, too, and in the middle ages, finger- 
reckoning was a recognized part of the necessary 
equipment of the calculator.1 

It is, perhaps, not strange that, in the outburst of 
enthusiasm attendant upon the introduction of the 
Hindu numerals in the schools of Western Europe, 
these mechanical aids should have been relegated to 
the curiosity shop. Neither is it strange to us, looking 
back, that there should have come a result quite un- 
foreseen by the educators of that time, namely, a loss 
of the power of real insight into number. Rules for 
computation existed and results were secured, but the 
realization of number was often sadly lacking. It was 
not until late in the eighteenth century that this loss 
was recognized and material aids to a comprehension 
of number were restored by Busse, Pestalozzi, and 
their associates. 

Arithmetic of the middle ages— Among pre-Chris- 
tian Europeans north of Italy we find little trace of 
arithmetical knowledge. At the beginning of our era 
learning was at a very low state throughout this region. 
Tacitus tells us that writing was unknown among the 
common people, although it was an accomplishment of 
‘the priests. As business increased, however, some 
mathematical knowledge became necessary even before 
our era. Salt and amber were exported from Central 


1 For description, see Gow, p. 24. 


HOW ARITHMETIC HAS DEVELOPED 59 


Europe, and Assyrian inscriptions tell of the purchase 
of the latter commodity from the North. Tacitus tells 
us that in his time the German tribes had come to 
know the Roman weights and coins, and hence they 
knew enough simple counting for trading purposes. 

To replace the primitive northern arithmetic, came, 
with the southern conquerors, the Roman. The domi- 
nant power soon made it to the financial interest of 
the traders to use the Italian numerals. And although 
Rome had done little for education, some of her later 
statesmen recognized the value of scholarship, as wit- 
ness Capella, Cassiodorus, and Boethius, and this fact 
made the northern tribes incline to education. Rome, 
however, had contributed so little that, when her power 
in the North declined, it was hardly to be expected 
that there should be any decided contribution to knowl- 
edge among her former subjects. Nevertheless, in 
Gaul, where the Franks established a well-ordered 
monarchy, schools were founded, and the French king, 
Chilperic (d. 584), devoted himself with earnestness 
to a system of public education. The Merovingian 
princes erected a kind of Court school, after the man- 
ner of the Romans, and thus were founded the Castle 
schools which were common throughout the middle 
ages. Naturally, however, these schools contributed 
nothing to mathematics; the training of a knight 
did not require the exact sciences. 


1 Sterner, p. IOI. 


60 THE TEACHING OF ELEMENTARY MATHEMATICS 


The Church schools did more for mathematics, as 
for learning in general. Wherever the Church went, 
there went the school. By whatever name known, 
_whether cloister, cathedral, or parochial, they existed 
in connection with every large ecclesiastical founda- 
tion. Especially did the schools of St. Benedict of 
Nursia,! starting from the parent monastery at Monte 
Cassino (near Naples), spread all over Western Europe, 
until the Benedictine foundations became the recog- 
nized centres of learning from the Mediterranean to 
the North Sea. 

In these Church schools mathematics had some little 
standing. The quadrivium of arithmetic, music, ge- 
ometry, and astronomy, was commonly recognized in 
higher education, and in spite of the low plane on 
which arithmetic was usually placed (see p. 59), some 


were found to assign it a worthy place. To Isidore, — 


to Bede the Venerable, to St. Boniface, to Alcuin of 
York, and other Church leaders, we owe the little 


standing that arithmetic had during the early middle 


ages. It was doubtless at Alcuin’s suggestion that 
Charlemagne decreed that the schools should ‘ make 


1 480-543. Called by Gregory the Great, “scienter nesciens, et sapi- 
enter indoctus,” learnedly ignorant and wisely unlearned. 

2 So Isidore of Seville, one of the most influential of medizeval writers, 
says: “Tolle numerum rebus omnibus et omnia pereunt. Adime seculo 
computum et cuncta ignorantia caeca complectitur, nec differi potest a 
ceteris animalibus qui calculi nescit rationem.’’— Origines, Lib. III, 


cap. 4, § 4. 


HOW ARITHMETIC HAS DEVELOPED 61 


no difference between the sons of serfs and of free 
men, so that they might come and sit on the same 
benches to study grammar, music, and arithmetic,” ! 
and that ‘‘the ecclesiastics should know enough of 
arithmetic and astronomy to be able to compute the 
time of Church festivals.” 2 

Brief reference has already (p. 5, 15) been made to 
the fact that men, being trained in the monasteries 
for ecclesiastical work, could get from arithmetic two 
things which correlated with their professional in- 
terests. One was the ability to compute the date of 
Easter (whence comes the chapter on the calendar), 
and the other was the training in disputation and in 
puzzling an opponent (whence come many inherited 
and useless puzzles of our arithmetics and algebras 
of to-day). A further example of these puzzles of 
Alcuin’s time may be of interest: ““Two men bought 
some swine for 100 solidi, at the rate of 5 swine for 2 
solidi. They divided the swine, sold them at the same 
rate at which they bought them, and yet received a 
profit. How could that happen?”? The puzzle is 
unravelled by seeing that the swine were of different 
values. There were 120 sold at 2 for 1 solidus, 120 at 
3 for 1 solidus, so that 5 went for 2 solidi as before; 
120 good ones therefore brought 60 solidi, and 120 


1 Capitularies of 789, art. 70; quoted by Guizot, History of France, I, 


p. 248. 
2 Sterner, p. IIo, 8 Cantor, I, p. 787; Sterner, p. IIo. 


62 THE TEACHING OF ELEMENTARY MATHEMATICS 


poorer ones 40 solidi, so the dealers had their 100 
solidi and still had 10 swine left by way of profit. | 

To weed out problems of this kind has taken a 
long time, and even the present generation finds now 
and then some advocate of exercises almost as absurd, 
as sharpeners of the wit. 

The period from Bede to the tenth century, one 
of the darkest of the middle ages, saw arithmetic 
largely given up to the computing of Easter, the com- 
putist becoming so prominent that the Germans have 
designated the period as that of the ‘‘Computists.” } 

Another movement of importance, to which allusion 
has already been made, followed this period of degen- 
eracy. The Hanseatic League, arising from a union 
of German merchants abroad and of their important 
commercial centres at home, attained its first prom- 
inence in the thirteenth century. Although it had for 
its primary object the protection of the trade routes 
between the allied cities, it soon developed other objects, 
such as the assertion of town independence against the 
rapacity of the feudal aristocracy, the establishment 
of warehouses along the paths of commerce, the formu- 
lation of laws of trade, and the general improvement 
of commercial intercourse. Among these acts was the 
establishment of the Rechenschulen (reckoning schools, 
arithmetic schools). The inadequacy of the business 
course in the Church schools, and the unsatisfactory 


1 Sterner, p. 115; but see Cantor, I, p. 783. 


HOW ARITHMETIC HAS DEVELOPED 63 


attempts at teaching bookkeeping, arithmetic, etc., led 
to the creation of the office of Rechenmeister already 
described. The guild of Rechenmeisters included some 
of the best teachers of the time, — Ulrich Wagner of 
Niirnberg, who wrote the first German arithmetic (1482), 
Christoff Rudolff, who wrote the first German algebra, 
Grammateus, who wrote the first German work on book- 
keeping, and others equally celebrated. So powerful 
did this monopoly become, that for a long time it kept 
arithmetic out of the common schools, and it is in part 
due to this influence that not until Pestalozzi’s time was 
arithmetic taught to children on entering school. 

When at last it was decided that arithmetic could 
profitably be taught in the earliest grades, the inherited 
work of the Rechenmeisters was dropped in upon the 
lower classes, and it is chiefly due to this fact that we 
have had, even to the present day, a mass of business 
problems (often representing customs of the days of 
the Rechenschulen, but long since obsolete, like part- 
nership involving time) in the fifth, sixth, and seventh 
| grades, where they are almost wholly unintelligible. 
| The period of the Renaissance— The period of the 





rebirth of learning, the Renaissance, is one of the most 
| interesting which the historian meets. Manifold causes 
contributed to make the close of the fifteenth century 
Jan era of remarkable mental activity. The fall of 
Constantinople (1453) turned the stream of Greek cul- 
\ture westward, and it reached the shores of Italy with 


64 THE TEACHING OF ELEMENTARY MATHEMATICS 


a power far in excess of that which it exerted in the 
region of the Bosphorus. Joined to this were the 
revelations of that new astronomy which, by the help 
of mathematics, was to overthrow the Ptolemaic theory ; 
the discovery of a new continent and the consequent 
revival of commerce; the invention of cheap paper and 
of movable type, two influences which gave wings to 
thought; and, not the least of all, that great movement 
known as the Reformation, which set men thinking as 
well as believing. From this period of migration, of 
discovery, of invention, and of independent thought, 
dates arithmetic as we know it. 

It is not difficult to see what would naturally find 
place in arithmetic at that time. Crystallized in the 
new printed works would be the arithmetic which the 
Greeks brought from Constantinople, — the theory of 
numbers and roots by geometric diagrams. The Roman 
numerals, which had been used almost exclusively to 
this time, would find a prominent place. The Arab 
arithmetic, coming in with the Hindu numerals (already 
more or less known), would contribute its little share 
in the way of alligation, Rule of Three (our simple 
proportion), and series, which last was known in 
classical times as well. 

Together with this inherited matter would naturally 
be placed the arithmetic demanded by the peculiar 
conditions of the time. The small states, with their 
diverse monetary systems, demanded an _ elaborate 


HOW ARITHMETIC HAS DEVELOPED 65 


method of exchange, not merely “simple,” but also 
“arbitrated.”” The absence of an elaborate banking 
system like that of to-day rendered the common draft 
one payable after, instead of at sight. The various 
systems of measures in the different states and cities 
required elaborate tables of denominate numbers,} 
and the lack of decimal fractions explains the need 
of compound numbers with several denominations. 
The frequent reductions from one table to another, 
necessitated by these circumstances, encouraged the 
use of the Rule of Three (Regula de tri, Regeldetri, 
‘Regula aurea), so that this piece of mechanism came 
to be esteemed quite highly in the arithmetics of 
that time. Then, too, the demands of commerce 
brought in problems in the mensuration of masts and 
sails, and those which finally developed in our Amer- 
ican text-books as General Average. Stock com- 
panies not having as yet been invented, elaborate 
problems in partnership, involving different periods 
of time, were a necessary preparation for business. 
Later, business customs demanded Equation of Pay- 
ments, a scheme not uncommon in days when long 
standing accounts were the fashion between whole- 
salers and retailers. Such were some of the condi- 
tions in the days when printing was crystallizing the 
science of arithmetic. 


1Thus Graffenried’s Arithmetica Logistica, 1619, has 21 pages of 
tables. 


66 THE TEACHING OF ELEMENTARY MATHEMATICS 


Arithmetic since the Renaissance — There have been 
several improvements in methods of calculating since 
the period of revival in Italy, and the business 
changes have revolutionized the commercial side of 
arithmetic. 

Among the improvements in pure arithmetic, the 
most important can be stated briefly. The first has 
to do with the invention of the common symbols of 
operation, which may, in a rough way, be placed in 
the century from 1550 to 1650.1 Prior to this time 
the statement of the operations was set forth in full, 
and for any material advance some stenography or 
symbolism was necessary. 

The second improvement relates to the invention 
of decimal fractions about 1600, an invention due 
perhaps as much to Biirgi as to any one.? But al- 
though these fractions appeared three centuries ago, 
it was not until about 1750 that they found much 
footing in the schools, so conservative are schoolmas- 
ters, their constituents, and the various examining 
authorities. With the establishment of the decimal 
fraction, however, arithmetic was revolutionized, per- 
centage became synonymous with advanced business 
calculations, the greatest common divisor (necessary 


1 A brief historical note upon the subject may be found in Beman and 
Smith’s Higher Arithmetic, Boston, 1896, p. 43. 

2 Stevin, Kepler, Pitiscus, and others had a hand in the invention. 
See Cantor, II, p. 567. 


HOW ARITHMETIC HAS DEVELOPED 67 


in the days of extensive common fractions) became 
obsolete for scientific purposes, and science found a 
new servant to assist in her vast computations. 

The third improvement is the invention of loga- 
rithms by Napier in 1614.1! One might expect that a 
scheme which, by means of a simple table, allowed 
computers to multiply and divide by mere addition 
and subtraction, would find immediate recognition in 
the schools. And yet, so conservative is the pro- 
fession that, even in high schools in English speak- 
ing countries, logarithms find almost no place, in 
spite of the fact that neither in theory nor in prac- 
tice do they present any difficulties commensurate 
with many found in the old-style arithmetic. In Ger- 
many the schools are more progressive in this matter. 

The fourth improvement of moment is seen in our 
modern methods of multiplication and division. A 
problem in division three hundred years ago was a 
serious matter. The old “scratch” or “galley” 
method? was cumbersome at the best, and the in- 
troduction of the ‘Italian Method,” which we com- 
monly use, was a great improvement. Nor is the 
day of change in these operations altogether passed, 


1 That is, his “ Descriptio mirifici logarithmorum canonis” appeared in 
that year. The best brief discussion of the relative claims of Napier and 
Biirgi is given in Cantor, II, p. 662 seq. 

2 Well illustrated in Brooks, E., Philosophy of Arithmetic, Lancaster, 
Pa., 1880, p. 55, 59. 


68 THE TEACHING OF ELEMENTARY MATHEMATICS 


for just now we have the “Austrian methods” of 
subtraction and of division coming to the front in 
Germany, and we may hope soon to see them com- 
monly used in the English-speaking world. 

The fifth improvement is partly algebraic. Algebra, 
as we know it with its present common symbolism, 
dates only from the early part of the seventeenth cen- 
tury. With its establishment there departed from arith- 
metic all reason for the continuance of such subjects as 
alligation (an awkward form for indeterminate equa- 
tions), series (better treated by algebra), roots by the 
Greek geometric process, Rule of Three (as an unex- 
plained rule), and, in general, the necessity for any 
mere mechanism. Mathematicians recognize no divid- 
ing line between school arithmetic and school algebra, 
and the simple equation, in algebraic form, throws such 
a flood of light into arithmetic that hardly any leading 
educator would now see the two separated. 

The present status of school arithmetic is one of 
unrest. We have these inheritances from the Renais- 
sance, and with difficulty we are breaking away from 
them. Only recently have we seen alligation disap- 
pear from our text-books, and slowly but surely are 
we driving out “true” discount, equation of payments, 
arbitrated exchange, troy and apothecaries’ measures, 
compound proportion, and other objectionable matter. 
Such subjects, are, as already suggested, unworthy of 
a place in the course which is to fit for general citi- 


HOW ARITHMETIC HAS DEVELOPED 69 


zenship; for they are practically obsolete (like troy 
weight), or useless (like arbitrated exchange), or mere 
mechanism and show of knowledge (like compound 
proportion), or they give a false idea of business (like 
“true” discount). 

Slowly we are opening the door to the simple equa- 
tion, because it illuminates the practical problems of 
arithmetic, especially those of percentage and propor- 
tion. “It is evident,” says M. Laisant, “that all 
through the course of arithmetic, letters should be 
introduced whenever their use facilitates the reasoning 
or search for solutions.” ? 

The present tendency is decidedly in favor of elimi- 
nating the obsolete, of substituting modern business for 
the ancient, of destroying the artificial barrier between 
arithmetic and algebra, and of shortening the course in 
applied arithmetic. As the report of the “ Committee 
of Ten’ stated the case, ‘‘ The conference recommends 
that the course in arithmetic be at the same time 
abridged and enriched; abridged by omitting entirely 
those subjects which perplex and exhaust the pupil 
without affording any really valuable mental discipline, 
and enriched by a greater number of exercises in simple 
calculation and in the solution of concrete problems.” ? 
Three years later, the ‘‘ Committee of Fifteen” had this 


1 La Mathématique, p. 206. 
2 For full report of the mathematical conference, see Bulletin No, 205, 
United States Bureau of Education, Washington, 1893, p. 104. 


70 THE TEACHING OF ELEMENTARY MATHEMATICS 


further suggestion: “Your Committee believes that, 
with the right methods, and a wise use of time in pre- 
paring the arithmetic lesson in and out of school, five 
years are sufficient for the study of mere arithmetic — 
the five years beginning with the second school year 
and ending with the close of the sixth year; and that 
the seventh and eighth years should be given to the 
algebraic method of dealing with those problems that 
involve difficulties in the transformation of quantitative 
indirect functions into numerical or direct quantitative 
data.’’} 

In all this present change and suggestion of change, 
the radical element in the profession is restrained by 
several forces: the publisher fears to join in a too 
pronounced departure; the author is also concerned 
with the financial result; the teacher is fearful of the 
failure of his pupils on some official examination (a 
most powerful influence in hindering progress); and 
the pupil and his parents see terrors in any depart- 
ure from established traditions. But in spite of all 
this, the improvement in the arithmetics in America 
has, within a few years, been very marked — more so 
than in any other country. 


1 Report of the Committee of Fifteen, Boston, 1895, p. 24. 


CHAPTER IV 
How ARITHMETIC HAS BEEN TAUGHT 


The value of the investigation of the way in which 
arithmetic has been taught, especially during the nine- 
teenth century, is apparent. Find the methods fol- 
lowed by the most successful teachers, find the failures 
made by those who have experimented on new lines, 
and the broad question of method is largely settled. 
“The science of education without the history of educa- 
tion is like a house without a foundation. The his- 
tory of education is itself the most complete and 
scientific of all systems of education.” } 

It is impossible at this time to trace the develop- 
ment of the general methods of teaching the subject, 
up to the opening of the nineteenth century. Already, 
in Chapter I, the development of the reasons for 
teaching the subject has been outlined, and from this 
the general methods employed may be _ inferred. 
Only a hurried glance at a few of the more interest- 
ing details is possible. 

The departure from object teaching — Arithmetic, 
at-least in the Western world, was always based upon 
object teaching until about 1500, when the Hindu 


1 Schmidt, Geschichte der Padagogik, I, p. 9. 
71 


72 THE TEACHING OF ELEMENTARY MATHEMATICS 


numerals came into general use. But in the enthu- 
siasm of the first use of these symbols, the Christian 
schools threw away their abacus and their numerical 
counters, and launched out into the use of Hindu 
figures. And while they saw that the old-style ob- 
jective work was unnecessary for calculation, which 
is true, they did not see that it was essential as a 
basis for the comprehension of number and for the 
development of the elementary tables of operation. 
Hence it came to pass that a praiseworthy revolution 
in arithmetic brought with it a blameworthy method 
of teaching. Although there were better tools for 
work —the Hindu numerals, arithmetic became even 
more mechanical than before, and it was not until the 
time of Pestalozzi, three centuries later, that educators 
awoke to the great mistake which had been made in 
discarding objects as a basis for number teaching. 
With the introduction of the Eastern figures, text- 
books became filled with rules for operations, and 
teachers followed books in this mechanical tendency. 
To define the terms, to learn the rules, to repeat the 
book, this was the almost universal method for three _ 
hundred years before Pestalozzi, and even yet the 
method has not entirely died out.! A modern math- 
1 Janicke and Schurig’s Geschichte der Methodik des Unterrichts in den 

mathematischen Lehrfachern, Band III of Kehr’s Geschichte der Meth- 
odik des deutschen Volksschulunterrichtes, Gotha, 1888. The first part 


of the volume is Janicke’s Geschichte der Methodik des Rechenun- 
terrichts, and will hereafter be referred to as /admicke. Janicke, p. 21. 


HOW ARITHMETIC HAS BEEN TAUGHT 73 


ematician would fare ill in passing an arithmetic ex- 
amination of those days, before their examiners, ! 
just as the mathematician of a couple of centuries 
hence will wonder at the absurdities of many of our 
questions to-day. 

Rhyming arithmetics — The difficulty of committing 
to memory a large number of rules upon the subject 
led educators to look for a remedy. Some, and 
among them Ascham and Locke, mildly protested 
against so many rules, but for a long time a large 
number was considered necessary, and this plan is 
even yet advocated by many teachers. Among the 
remedies suggested was that of putting the rules in 
thyme, the argument being that (1) a multitude of rules 
is a necessity, (2) rhymes are easily memorized, (3) 
hence this multitude of rules should appear in rhyme, 
—a good enough syllogism if we admit the major 
premise. Hence for a long time rhyming rules were in 
vogue, and might be to-day had not opinions changed 
as to the value of the rule itself. Even during the 
last quarter of the nineteenth century, however, an 
arithmetic in rhyme appeared in New York State — 
so little aré the lessons of the history of methods 
known. 

Form instead of substance was a natural outcome 
of the policy of making arithmetic purely mechanical. 
So we find much attention paid to the preparation of 


1 For such a paper see Janicke, p. 22. 


74. THE TEACHING OF ELEMENTARY MATHEMATICS 


artistic copybooks with curious arrangements of work. 
The following may serve to illustrate the results of 
this tendency :! 





79745 97548 
54789 69457 
30 63 
2420 48 
361635 4549 
54242840 2472 
42360423245 363535 
28634836 303632 
497254 81282528 
5681 42451640 
63 5463202056 
5 160119905 6775391436 


It is possible that to this tendency to prepare artis- 
tic copybooks rather than to acquire facility in arith- 
metic there is to be attributed the continued use of 
the old “scratch” or “galley”? method of division, 
long after the more modern Italian method was 
known. 

Instruction in method, for teachers of arithmetic, 
began to appear in noteworthy form about the mid- 
dle of the seventeenth century, “like an oasis in a 


1 Janicke, p. 27. 
2 This method is given in all of the histories of mathematics already 
named. 


HOW ARITHMETIC HAS BEEN TAUGHT 75 


still 
unlimited number space, then addition in such space, 


desert,” says Janicke. But the plans suggested were 
echanical — counting and writing numbers in 


then subtraction, and so on. “The teacher,” says 
one of the best works of the time, ‘is to write the 
first nine numbers, then pronounce them four or five 
times, then let the boys, one after another, repeat 
them.” 

A picture of the best methods employed at the 
opening of the eighteenth century may be seen in 
the rules for the celebrated Francke Institute at 
Halle (1702),! rules not without suggestiveness to cer- 
tain teachers to-day: 

“All children who can read shall study arithme- 


”» 


tic.” It was not until about a century later that the 
subject was taught to children just entering school, 
and to-day we have quite a pre-Pestalozzian move- 
ment to the old plan, akin to pre-Raphaelitism in the 
graphic arts. 

“On account of the diverse aptitudes of children, 
in the matter of arithmetic, it is impossible to form 
classes; hence the teacher shall use a printed book 
and shall teach the subject from it.... He shall 


go around among the children and give help where 


1 Unger, p. 140; Janicke, p. 32. In general it may be said that any 
one who wishes to follow the development of method in arithmetic must 
consult these works. There is nothing more systematic than Unger, 
nothing so complete as Janicke. 


76 THE TEACHING OF ELEMENTARY MATHEMATICS 


it is necessary.” To-day we hear not a little of ‘“‘the 
laboratory method” and “individual teaching,” a re- 
turn to the methods of the past, methods in which 
the inspiration of community work was wanting, 
methods long since weighed in the balance and 
found wanting. 

“The teacher must dictate no examples, but each 
child shall copy the problems from the book and 
work them out in silence.’ This plan is also not 
unknown in the teaching of the subject to-day. 

“Tt would be a good thing if the teacher would 
himself work through (durchrechnet) the book so that 
he could help the children” ! 

It was toward the close of the eighteenth century 
that the modern treatment of elementary arithmetic 
began to show itself. In the Philanthropin at Dessau, 
an institution to which education owes not a little, 
we find in 1776 very little improvement upon the old 
plan of pretending to teach all of counting, then all 
of addition, then all of subtraction, and so on.! But 
in the following year Christian Trapp began upon 
entirely new lines, and in 1780 he published his ‘“ Ver- 
such einer Padagogik,” in which he worked out quite 
a scheme of teaching young children how to add and 
subtract, objects being employed and the effort being 
made to teach numbers rather than figures. This he 
followed by simple work in multiplication and division, 


1 Janicke, p. 44. 


HOW ARITHMETIC HAS BEEN TAUGHT viv 


and he worked out a systematic use of a box of blocks 
illustrating the relation of tens to units, a forerunner 
of the Tillich reckoning-chest mentioned later! It is 
here that we may say, with fair approximation to justice, 
the modern teaching of elementary arithmetic begins. 
Trapp’s successor was Gottlieb von Busse, whose first 
works on arithmetic appeared in 1786. He was still 
wedded to the old system of first teaching numera- 
tion (to trillions), then the four fundamental processes 
in order, and so on. But at the same time he made 
a distinct advance in the systematic use of © e e 
number pictures (Zahlenbilder, translated by ° ° 
3) 
being associated with the group as here five 


some genius as “number duz/ders’’/), points 


shown. He used special forms for tens (to dis- 
tinguish them from the unit dots), and also for the 
hundreds and the thousands, thus carrying a good 
thing to a ridiculous extreme.” In the same way we 
still have in our day not a few failures as a result 
of carrying objective teaching too far. This is one 
of Grube’s errors, although few would follow him 
closely enough to be harmed by it. 

Mentron should also be made of the work of a 
nobleman, Freiherr von Rochow, of Rekan, near 
Brandenburg, who is known as the reformer of the 
country schools of Germany,? and whose influence led 


1 Janicke, p. 44. 2 Ib., p. 45 seq. ; Unger, p. 165 seq. 
* Unger, p. 138. 


78 THE TEACHING OF ELEMENTARY MATHEMATICS 


to the attempt on the part of his assistants to make 
arithmetic attractive instead of insufferably dull, and 
to use it for training the mind as well as for a prepa- 
ration for trade. 

Pestalozzi— Trapp, Busse, von Rochow, and a few 
others whose names and work can hardly be men- 
tioned here, were like “the voice of one crying in 
the wilderness’”’; there was another who should come. 
Johann Heinrich Pestalozzi, a poor Swiss schoolmaster, 
a man who seemed to make a failure of whatever he 
undertook, laid the real foundation of primary arith- 
metic as it has since been recognized. He wrote no 
work directly upon the subject, and one who searches 
for his ideas upon number teaching has to pick a 
little here anda little there from among his numerous 
papers and letters, and take the testimony of those 
who knew him.? | 

Number had been taught to children by the aid 
of objects before Pestalozzi began his work. This, 
indeed, as already stated, was the primitive plan, amd 
was thrown over only with the introduction of print- 
ing and the Hindu numerals.! Trapp and Busse had 
tried, not to revive the old plan of using’ objects 
for all calculations, but to make a reasonable use of 
objects with beginners. /Their plans were crude, 
however, and it was reserved for Pestalozzi scientif- 


1 Janicke, p. 48, 46. 2 Ib., p. 63; Unger, p.-176, 


HOW ARITHMETIC HAS BEEN TAUGHT 79 


ically to make perception the basis for all number 
work.! 

Of course this does not mean that Pestalozzi was 
the first to recognize the value of perception. This 
was not at all new. The ancients understood it 
well, and Horace even placed it in his verse: “The 
things which enter by the ear affect the mind more 
languidly than such as are submitted to the faithful 
dges.” ? 

Pestalozzi, however, was the first to recognize its 
value to the full, and to put it to practical use in 
teaching.® 

With Pestalozzi, too, the formal culture value of 
number came definitely and systematically to the front, 
the value of “mental gymnastic” (Geistesgymnastik) 
was recognized —unduly so, to be sure, and all daw- 
dling “busy work” was wanting. The children worked 
rapidly, cheerfully, orally. They showed themselves 
quick in number work, wide awake, active, and we can 
learn more to-day from Pestalozzi than from any other 
‘one teacher of the subject, and this in spite of all the 
‘faults of method which he unquestionably possessed. 


| 


1“Die Anschauung ist das absolute Fundament aller Erkenntniss.” 
_—-Pestalozzi to Gessner. Compare Diesterweg: “ Das ganze Geheimniss 
|der Elementarmethode ruht in der Anschaulichkeit.” 
2«Segnius irritant animos dimissa per aurem, 

Quam quae sunt oculis subiecta fidelibus.”— Ars poetica, v. 180. 
| 8 Shafer, Fr., Geschichte des Anschauungsunterrichts, in Kehr’s 
‘Geschichte der Methodik, I, p. 468. 


| 


80 THE TEACHING OF ELEMENTARY MATHEMATICS 


It is related of him! that a Niirnberg merchant, who 
had heard with some doubts of his success in teaching 
arithmetic, came to the school one day and asked to 
be allowed to question the boys. The request being 
granted, he proposed a rather complicated business 
problem involving fractions. To his astonishment the 
boys inquired whether he wished it solved in writing 
or ‘in the head,” and upon his naming the latter plan 
he began for himself to figure out the result on paper; 
but before he had half done the boys’ answers began 
to come in, so that he left with the remark, “I have 
three youngsters at home, and each one shall come 
to you as soon as I can get there.”* The incident, 
possibly exaggerated, is not unique; Biber® and others 
relate numerous instances of the success which at- 
tended Pestalozzi’s earnest work in oral arithmetic 
founded upon perception. 

Pestalozzi was not narrow in his ideas as to the 
objects to be employed, as Tillich and many other 
teachers of later times have been. This particular 
device (say some form of abacus), or that (as some 
set of cubes, or disks, or other geometric forms), did 
not appeal to him. He used, to be sure, an arrange- 


1 By Blockmann, “ Heinrich Pestalozzi, Ziige aus dem Bilde seines — 
Lebens,.” Dresden, 1846. 

2 See also De Guimp’s Pestalozzi, American ed., p. 214. 

3 Life of Pestalozzi, p. 227 et pass. It is unfortunate that this ex- ~ 
cellent work has become so rare. | 


HOW ARITHMETIC HAS BEEN TAUGHT 81 


ment of marks on a chart (his “ units’ table,” Einheits- 
tabelle), but he did not limit himself to any such 
device; he led the child to consider all objects which 
were of interest to him, nor did he fear (O modern 
teacher!) to let him use the most natural calculating 
device of all—the fingers.} 

Pestalozzi’s leading contributions may be summed up 
as follows: 

1. He taught arithmetic to children when they first 
came to school, basing his work upon perception, and 
seeking to make the child independent of all rules 
and traditions. Nevertheless, he did not wholly free 
the subject from mechanism. He avoided the baser 
form which depended upon rules and principles, but 
he substituted a mechanism of forms based upon per- 
Sepuon, lis never ending 2x I +3 x = 2x 174s 
very tiresome in spite of its value for ‘beginners.? 

2. He insisted that the knowledge of number should 
precede the knowledge of figures (Hindu numerals), in 
the number space from 1 to 10. ‘“ Now it is,” said he, 
“a matter of great importance that this ultimate basis 
of all number should not be obscured in the mind by 


1 The best insight into Pestalozzi’s ideas along this line is given in 
the work of his friend and co-worker, Kriisi, Anschauungslehre der 
Zahlenverhaltnisse, Zurich, 1803. 

2 Damit fiihrte er in der Darbietung vom vorpestalozzischen puren 
Mechanismus zum anschaulichen Zahlmechanismus, an dem unser ele- 
. mentarer Rechenunterricht auch heute noch krankt.” Brautigam, 
| Methodik des Rechen-Unterrichts, 2. Aufl, Wien, 1895, p. 2. 


G : 


82 THE TEACHING OF ELEMENTARY MATHEMATICS 


arithmetical abbreviations.” ! Tillich, Pestalozzi’s most 
talented follower, agrees with his master in this. “The 
figures,” he writes, ‘‘are only the symbols for numbers. 
Hence they ought not to be taught to the child until 
the numbers are familiar to him.’ To do otherwise is 
to make the same mistake that one would make in 
teaching letters to a child who could not yet talk,’ a 
rather radical statement, but one with a core of truth. 
First and foremost the child must conceive of xumder; 
figures, operations, applications beyond mere counting 
and selecting of groups, these could wait. As one of 
the modern opponents of Grube’s heresy has put it, 
“First the number concept, then the operations.” 3 

3. He also insisted that the child should know 
the elementary operations before he was taught the 
Hindu numerals. ‘‘When a child has been exercised 
in this intuitive method of calculation as far as these 
tables go (z.e. from 1 to 10), he will have acquired 
so complete a knowledge of the real properties and 
proportions of number as will enable him to enter 
with the utmost facility upon the common abridged 
methods of calculating by the help of ciphers.’’4 

4. The Hindu numerals followed this training in 
pure number. “His mind is above confusion and 


1 Letter to Gessner, Biber’s Pestalozzi, p. 278. 
2 Lehrbuch der Arithmetik, p. 41. 

8 Beetz, K. O., Das Wesen der Zahl, p. 204. 
4 Letter to Gessner, Biber’s Pestalozzi, p. 282. 


HOW ARITHMETIC HAS BEEN TAUGHT 83 


trifling guesswork; his arithmetic is a rational pro- 
cess, not a mere memory work or mechanical routine; 
it is the result of a distinct and intuitive apprehen- 
sion of xumber.” } 

5. Fractions were treated in the same way; first 
the concept of fraction, then some exercise in opera- 
tions, finally the shorthand characters. After the 
child has “such an intuitive knowledge of the real 
proportions of the different fractions, it is a very 
easy task to introduce him to the use of ciphers for 
fraction work.”? After all, Pestalozzi was simply 
following out Ratke’s well-known rule, “First a 
thing in itself, and then the way of it; matter before 
form.” The only -question is, Did he postpone the 
form too long? 

6. He made arithmetic the most prominent study 
in the curriculum. “Sound and form often and in 
various ways bear the seeds of error and deceit; 
number never; it alone leads to positive results.” ® 
“T made the remark,” said Pére Girard, himself one 
of the foremost Swiss educators, “to my old friend 
Pestalozzi, that the mathematics exercised an unjusti- 
fiable sway in his establishment, and that I feared 
the results of this on the education that was given. 
Whereupon he replied to me with spirit, as was his 
manner, ‘This is because I wish my children to 


1 Letter to Gessner, Biber’s Pestalozzi, p. 282. 3 Tb., p. 283. 
8 Pestalozzi’s Simmtliche Werken, 11. Bd., p. 226. 


84 THE TEACHING OF ELEMENTARY MATHEMATICS 


believe nothing which cannot be demonstrated as 
clearly to them as that two and two make four.’ 
My reply was in the same strain: ‘In that case, if 
I had thirty sons, I would not intrust one of them 
to you, for it would be impossible for you to dem- 
onstrate to him, as you can that two and two 
make four, that I am his father, and that I have a 
right to his obedience.’”! Thus did Pestalozzi give 
to arithmetic an exaggerated value (not that the Pére’s 
argument is very convincing), and thus it assumed a 
prominence in the curriculum which his followers 
maintained, and which is only now, after the lapse of 
a century, being questioned by leading educators. 

7. He emphasized oral arithmetic as a mental 
gymnastic, but he unquestionably carried the exer- 
cises too far. Knilling, who in his first work wrote 
with more force than judgment, was not wide of the 
mark when he said: “The exercises with Pestalozzi’s 
Rechentafeln and Einheitstabelle (number and units’ 
tables) belong to the most monstrous, most bizarre, 
most extravagant, and most curious that have ever 
appeared in the realm of teaching.’’? 


1 Payne’s trans. of Compayré’s History of Pedagogy, p. 437. 

2 Zur Reform des Rechenunterrichtes, I, p. 58. Those who care to 
know the weak points of Pestalozzi, Grube, and other German JZetho- 
dikers, and to find them discussed in vigorous language, should read this 
work. The later and more valuable works by the same author are also 
worthy of study: Die naturgemadsse Methode des Rechen-Unterrichts in 
der deutschen Volksschule, I. Teil, Miinchen, 1897; II. Teil, 1899. 


HOW ARITHMETIC HAS BEEN TAUGHT 85 


8. He abandoned the mechanism of the old cipher- 
reckoning, just as, three centuries before, the cipher- 
reckoners (algorismists) had abandoned the abacus, and 
put oral arithmetic to the front. Number rather than 
figures, was his cry. But while instituting a healthy 
reaction against the mechanical rules of his predeces- 
sors, like most reformers he went to the other extreme, 
so much so that the art of ciphering became quite 
distinct from his arithmetic. Against this extreme in 
due time another reaction set in and, in America, drove 
out the ‘“‘mental arithmetic,” which Colburn had done 
so much to establish, replacing it by the worst form 
of mechanism. In turn, against this movement another 
reaction has set in, and the close of the nineteenth 
century is seeing arithmetic beginning to be placed 
upon a much more satisfactory foundation than ever 
before. 

Of Pestalozzi’s contributions to arithmetic but two 
seriously influenced the world, perception as the foun- 
dation of number teaching, and formal culture as the 
aim. Although the creator of a method, it found little 
general recognition in Germany, and it is known to-day 
almost only by name.! 

1 Hoose’s Pestalozzian Arithmetic, Syracuse, 1882, made the method 
known, in its most presentable form, to American teachers. The bibliog- 
raphy relating to Pestalozzi is so extensive that it is hardly worth 
attempting to mention it. A brief résumé of his work is given in Com- 


payré’s History of Pedagogy, and generally in works of similar nature. 
Janicke gives the most judicial summary of the conflicting views con- 


86 THE TEACHING OF ELEMENTARY MATHEMATICS 


Tillich — Pestalozzi had a host of followers among 
writers even though his own method found little favor 
with teachers. Among the first of the prominent ones 
was Tillich,! who took for his motto the well-known 
but untranslatable words, ‘‘Denkend rechnen und 
rechnend denken,’” words which might be put into 
English as: “thinkingly to mathematize and mathe- 
matically to think.” Acknowledging the inspiring in- 
fluence of his master,? he nevertheless saw the faults 
of the latter's system and boldly attempted to rectify 
them. His plan may briefly be summed up as follows: 

1. He paid much attention to a systematic mastery 
of the first decade of numbers, making this the basis 
for the advanced work. “My method teaches one to 
know all possible relations in the first order (in the 
number space I-10), and by this means to form a stand- 
ard (eine Norm bilden) by which all higher numbers 
can be treated.” 

2. He did not attempt to bring a child to think of a 
number, 85 for instance, as so many units, but rather as 
cerning his theories. Knilling is the most interesting of his recent critics, 
especially in his first work, Zur Reform des Rechenunterrichtes, 1884; “I 
will,” he says, “make it as clear as day that all the modern errors in the 
teaching of primary arithmetic take themselves back to Pestalozzi,’ —I, 
p. 2. On the other hand, J. Riiefli is Knilling’s most interesting critic, in 
his work, Pestalozzi’s Rechenmethodische Grundsatze im Lichte der Kri- 
tik, Bern, 1890. 

1 Allgemeines Lehrbuch der Arithmetik, oder Anleitung zur Rechen- 


kunst fiir Jedermann, 1806. 
2 « Sein Feuer hat mich entflammt.” 


HOW ARITHMETIC HAS BEEN TAUGHT 87 


so many tens and so many units, and similarly for larger 
numbers, — a distinct advance on Pestalozzi, who failed 
to bring out the significance of the decimal system. 

3. To bring out prominently this relation between 
tens and units, and between the various units in the 
first decade, Tillich devised what he called a Reckon- 
ing-chest, a box containing 10 one-inch cubes, 10 paral- 
lelepipeds 2 inches high and an inch square on the 
base, 10 three inches high, and so on up to Io ten 
inches high. The use to which these rods were put 
is apparent, and it is also evident that the ratio idea 
of number was prominent in Tillich’s mind.t 

Of the other followers of Pestalozzi, space permits 
mention of only two. Tiirk? makes much of exercise 
in thinking, the formal training,? and follows Pesta- 
lozzi in taking up arithmetic first without the figures 
(in the number space I-20), but he departs from the 
plan of his master in not having the child begin the 
subject until his tenth year. The formal culture idea 
reached its height in the works of Kawerau;‘ his 
extreme views provoked the reaction. 


1 For a modern treatment of the subject see Braiutigam’s Methodik des 
Rechen-Unterrichts, 2. Aufl., Wien, 1895, p. 4 seq. 

2 Leitfaden zur zweckmissigen Behandlung des Unterrichts im Rech- 
nen, Berlin, 1816. 

3 Uebung im Denken, die Entwickelung und Starkung des Denkver- 
mogens, 

4 Leitfaden fiir den Unterricht im Rechnen nach Pestalozzischen Grund- 
satzen, Bunzlau, 1818. 


88 THE TEACHING OF ELEMENTARY MATHEMATICS 


Reaction against Pestalozzianism —It was natural 
that protests should arise against the extreme views of 
Pestalozzi and his followers. Like all reformers they 
were often intemperate in their demands and injudicious 
in their plans for improvement. The reaction was 
bound to come, and it was led by men of eminence in 
educational affairs, men to whom we are not a little 
indebted for certain opinions now generally held. 

For example, it was Friedrich Kranckes, whose first 
work appeared in 1819, who suggested the four concen- 
tric circles which Grube afterward adopted, exercising 
the child in the number space I-10, then in the space 
I-100, then I—1000, and finally I-10,000. He, as Busse 
had done before him, employed number pictures, and 
being one of the best teachers in North Germany, 
his influence greatly extended their use. He called his 
plan the Method of Discovery (Erfindungsmethode), — 
and developed his rules from exercise and observation. 
His problems, moreover, were not of the abstract 
Pestalozzian type; they touched the daily life of the 
child and avoided the endless formalism of the Swiss 
master. Such common-sense and sympathetic methods 
did not fail to win favor against Pestalozzi’s frag- 
mentary method. 

Denzel! was another master of the moderate school. 
He laid down these three aims in the teaching of pri- 


mary arithmetic: 


1 Der Zahlunterricht, Stuttgart, 1828, 


HOW ARITHMETIC HAS BEEN TAUGHT 89 


1. To exercise the thought, perception, memory ; 

2. To lead the children to the essence and the simple 
relations of number ; 

3. To give the children readiness in applying this 
knowledge to the concrete problems of daily life. 

This is a systematic and terse summary, and the third 
point is not one which played any part in the Pestaloz- 
zian scheme. Denzel, too, followed a concentric circle 
plan, treating the four operations in the circle I-10, 
then again in the circle I-20, and so on. 

Among the leaders who did the most to establish 
this moderate and common-sense school of teachers 
must be mentioned Diesterweg! and Hentschel,? men 
whose opinions have done much to mould the edu- 
cational thought of the last half century. 

Grube (1816-1884)?— Grube’s claim to rank as an 
educator lies largely in his power of judicious selection 
from the writings of others. He used the “concentric 
circle”’ notion, but this was half a century old; he 
made much of objective work, but so had every one 
since Pestalozzi; he insisted that ‘every lesson in arith- 
metic must be a lesson in language as well,’ but so 
had Pestalozzi. He gave, however, one new principle, 
—an extremely doubtful one,— that the four funda- 

1 Methodisches Handbuch fiir den Gesammturterricht im Rechnen, 
Elberfeld, 1829. 

2 Lehrbuch des Rechenunterrichtes in Volksschulen, 1842. 


8 Leitfaden fiir das Rechnen, Berlin, 1842. Trans. by Seeley (1891), 
and by Soldan (1878). 


90 THE TEACHING OF ELEMENTARY MATHEMATICS 


mental processes should be taught with each number 
before the next number was taken up,! and this is the 
essence, the only original feature, of the Grube method. 

The book was happily written; it was brief —not a 
common virtue; it was easily translated, and it thus be- 
came, some years ago, almost the only German “method” 
known in America. Thus it has come about that Grube 
has been looked upon as a name to conjure by, and 
neither the faults nor the virtues (much less the origi- 
nality) of the system seem to have been well considered 
by most of those who claim to use it,—claim to, for 
nobody actually does. 

Its chief virtue lies in its thoroughness. More than 
a year is given to the number space I-10, and three 
years are recommended for the space 1-100. Speak- 
ing of the number space I-10 he says: “In the 
thorough way in which I wish arithmetic taught, one 
year is not too long for this important part of the 
work. In regard to extent the pupil has not, appar- 
ently, gained very much; he knows only the numbers 
from 1 to 10,—but he knows them.” There is, how- 
ever, such a thing as being too thorough; to know 
all that there is about a number before advancing to 
the next one is as unnecessary as it is illogical, as 


1 Allseitige Zahlenbehandlung. 
2See the 6th (last) edition of the Leitfaden, 1881, p. 25, n.: “ Al- 
ways from the educational standpoint one must extend the first course 


a 


(z.e., I-100) over three years for the majority of pupils.” we 


HOW ARITHMETIC HAS BEEN TAUGHT OI 


impossible as it is uninteresting. Instead of requir- 
ing more time for the group I-10 when he published 
his sixth edition (1881) than he did when he pub- 
lished the first (1842), Grube might well have re- 
quired less. Home training and the training of the 
street are such that children know more about num- 
bers now than they did in the first half of the cen- 
tury. The interesting studies of Hartmann, Tanck, 
and Stanley Hall have shown that most children have 
a very fair knowledge of numbers to five before en- 
tering school. On the other hand, of course the ability 
to count must not be interpreted to mean that the child 
has necessarily any clear notion of number. Children 
often count to 100, as their elders often read poetry, 
with little attention to or appreciation of the meaning. 

The chief defects of the system are these: 

1. It carries objective illustration to an extreme, 
studying numbers by the aid of objects for three 
years, until 100 is reached.1 

2. It attempts to master each number before tak- 
ing up the next, as if it were a matter of importance 
to know the factors of 51 before the child knows 
anything of 75, or as if it were possible to keep 
children studying 4 when the majority know some- 
thing of 8 before they enter school. 

3. It attempts to treat the four processes simulta- 


1On the proper transition from the concrete to the abstract, see 
Payne’s trans. of Compayré’s Lectures on Pedagogy, p. 384. 


Q2 THE TEACHING OF ELEMENTARY MATHEMATICS 


neously, as if they were of equal importance or of 
equal difficulty, which they are not. 

While all must recognize that Grube gives many val- 
uable suggestions to teachers, the system as set forth 
in the last edition of the Leztfaden has almost no sup- 


porters. “While stimulating to children if not carried 


to excess, it easily degenerates into mere mechanism, as’ 


every one will agree who has carefully looked into it.” 7 
Of the later “methods,” but two or three can be 


mentioned. Kaselitz? has criticised his predecessors 
by saying that they teach a great deal about number, 
but do not teach the child how to operate wth num- 
ber. He therefore develops, and with much skill, the 
idea of making the number the operator. 

Knilling? and Tanck* are leaders in the modern 


1 Dittes, Methodik der Volksschule, 205. ‘Ein Instrument mit dem 


nur Meister umgehen kénnen.” — Bartholomai. ‘‘ Unmdglich, langweilig, 
zeitraubend, und ganz unniitz.... Die Behandlung jeder Einzelzahl ist 
unméglich und auch vdllig unniitz.” — Kallas, Die Methodik des elemen- 


taren Rechenunterrichts, Mitau, 1889, p. 20, 22. A good summary of the 
system is given in Unger, p. 188-195. An earnest protest against the 
whole system is set forth in Zwei Abhandlungen iiber den Rechenunter- 
richt, by Christian Harms, Oldenburg, 1889. The method is known to 
American teachers through translations of the earlier editions, made by 
Soldan and by Seeley. 

* Weegweiser fiir den Rechenunterricht in deutschen Schulen, Berlin, 
1878, and other works. 

8 Works already cited. For brief review see Hoffmann’s Zeitschrift, 
XXVIII. Jahrg., p. 514. 

# Rechnen auf der Unterstufe, 1884; Der Zahlenkreis von 1 bis 20, 
Meldorf, 1887 ; Betrachtungen iiber das Zahlen, Meldorf, 1890. 


HOW ARITHMETIC HAS BEEN TAUGHT 93 


pre-Pestalozzian movement. They assert that from 
Pestalozzi to the present time teachers have been 
assuming that number is the subject of sense-percep- 
tion, which it is not. ‘Number is not (psychologi- 
cally) got from things, it is put zzto them.”! They 
proceed to base their system upon the counting of 
things, a process in which three ideas are prominent, 
(1) the counted mass, (2) the how many, (3) the 
sense in which the things are considered. Knilling? 
classifies the numbers of arithmetic as (1) numbers 
of natural units—as of things, men, trees, etc.; (2) 
numbers of measured units—as of metres, grammes, 
etc.; (3) numbers of mathematical units. The mathe- 
matical unit is without quality (color, form, etc.); it 
is without extent; it is indivisible, a notion going 
back to Aristotle; it occupies no space; it is not 
imageable. Such a unit does not exist in the external 
world; it exists only in the mind. 

The child likes to count; the rhythm of counting 
is pleasing. “The fact that at least nearly all chil- 
dren, no matter how taught, first learn to count in- 
dependently of objects, in which the series idea gets 
ahead, — that they recognize three or four objects at 
first as individuals, calling the fourth one four even 
when set aside by itself, —that counting proceeds in- 


1 McLellan and Dewey, The Psychology of Number, p. 61. 
2 Die naturgemiasse Methodik, I, p. 55. 
8 Phillips, D. E., Pedagogical Seminary, V, p. 233. 


94 THE TEACHING OF ELEMENTARY MATHEMATICS 


dependently of the order of number names, and often 
consists in a repetition of a few names as a means 
of following the series,—that children desire and 
learn these names, —such, taken with the earlier 
steps presented, furnish unmistakable evidence that 
the series idea has become an abstract conception. 
The naming of the series generally goes in 
advance of its application to things, and the ten- 
dency of modern pedagogy has been to reverse this. 
Counting is fundamental, and counting that is 
spontaneous, free from sensible observation and from 
the strain of reason. ... In the application of the 
series to things is where the child first encounters 
much difficulty, and this is much increased because 
the teacher, not apprehending the full importance of 
this step, tries to hurry the child over this point 
entirely too rapidly. It is here that we meet 
with so many systems and devices for teaching 
numbers.” 1 | 
Upon this natural desire to count, Knilling and 
Tanck base their method, a systematic arrangement 
of counting forward and backward by ones, twos, 
etc., within the first hundred, leading easily to rapid 
work in addition, subtraction, multiplication, and even 
division. Mental pictures of numbers are of no value 
in actual work; all calculation is figure work; the 
head is never more empty of mental pictures than 


1 Phillips, D. E., Pedagogical Seminary, V, p. 221. 


HOW ARITHMETIC HAS BEEN TAUGHT 95 


when we calculate; calculation is not a matter of 
perception, it is a mechanical affair pure and simple. 

But given these exercises in running up and down 
the numerical scale, one is no nearer being an arith- 
metician than is one who can finger the scales on the 
piano to being a musician. Each furnishes the best 
basis for subsequent work and skill. 

One of the most temperate of writers upon this phase 
of number work? thus summarizes the discussion : 

I. Since through language number space was first 
created, and since here lies the source of all com- 
putation, therefore the teacher must impress upon the 
child the sequence of number words as a true, ser- 
viceable and lasting sound series (Lautreihe). 

2. Since with this series must in due time be asso- 
ciated things, perception enters. 

3. Since the number words establish only the chron- 
ological difference in the appearance of the individual 
units, suitable exercises should be given to make the 
pupil certain as to his order of the units. 

| This relation of number to time (sequence) is not 
new, and the subject has been a ground for debate 
since Kant first made it prominent. Sir William 
Hamilton takes one side and talks about “the science 
a pure time.” MHerbart® on the other hand main- 





1 “Diese Uebungen sind so wenig das Rechnen selbst, als Uebungen in 
en Scalen und in den Intervallen die Musik sind.” Fitzga, Di 23. 
_ ? Fahrmann, K. Emil, Das rhythmische Zahlen, Plauen i. V, 1896, p. 24. 
_ 3 Psychologie als Wissenschaft, TI, p. 162. 


Qg6 THE TEACHING OF ELEMENTARY MATHEMATICS 


tains that number is no more related to time than to 
a hundred other concepts. Lange relates number to 
space rather than to time, saying, “The oldest ex- 
pressions for number words relate, so far as we know 
their meaning, to objects in space.... The alge- 
braic axioms, like the geometric, refer to space-per- 
ceptions.”! ‘Every number concept is originally the 
mental picture of a group of objects, be they fingers 
or the buttons of an abacus.”? On the other hand, 
Tillich, whose method does not wholly agree with his 
sentiment, thus sets forth his views upon this point: 
“The empirical of arithmetic is to be sought in Time 
alone. It is therefore only the number arrangement 
which is capable of representation to the senses, and 
only the sequence which must be fixed in the first 
exercises, for from this everything else develops... . 
Number has nothing spatial about it, it exists only in 
Time, and not as anything absolute there, but only 
as something relative. The sequence is the great 
thing, not the magnitude.” 3 

This return to the pre-Pestalozzian idea of begin- 
ning with exercises in counting— but in a much more 
systematic way than any of Pestalozzi’s predecessors 
followed —is the latest phase of instruction in arith- 
metic which has commanded very general attention. 


1 Logische Studien, p. 140. 
2 Geschichte des Materialismus, II, p. 26. 
8 Lehrbuch der Arithmetik, p. 331, 333. 


HOW ARITHMETIC HAS BEEN TAUGHT 97 


The idea has been presented in America by Phillips.1 
But in working out the method in detail, the German 
writers have gone to an extreme, assigning “alto- 
gether too much value to counting —and to counting 
in a narrow sense, mere memory work with the num- 
ber series without reference to real things. ... It 
is a great overrating of the value of counting.... 
Counting should be the servant of number work, not 
number work the servant of counting.” ? 


1 Some Remarks on Number and its Applications, Clark University 
Monograph, 1898; Number and its Applications Psychologically consid- 
ered, Pedagogical Seminary, October, 1897. 

2 Grass, J., Die Veranschaulichung beim grundlegenden Rechnen, 
Miinchen, 1896, p. Io. 


CHAPTER V 
Tue PRESENT TEACHING OF ARITHMETIC 


Objects aimed at—In Chapter IV the growth of 
the teaching of primary arithmetic was briefly traced. 
The teaching of the more advanced portions was not 
considered. In the present chapter a few of the recent 
tendencies in both primary and secondary arithmetic 
will be briefly mentioned, and chiefly with a view to 
ascertaining what are a few of the points of con- 
troversy. 

In the first place, it is not at all settled as to what 
we are seeking in teaching arithmetic to a child. 
Herbart and his followers would have us bring out 
the ethical value. Others equally prominent and more 
numerous assert that it has no such value. “We en- 
tirely overrate arithmetic if we ascribe to it any 
soul-forming ethical power. ... The mental activity 
(Denkthatigkeit) induced by arithmetic is unproduc- 
tive and heartless (gemiitlos).’1 Grube and many. 
others would make it adapt itself to language work, 
Pestalozzi made much of the logical training which 
it gave, and several writers have amused themselves 


1 Korner, Geschichte der Padagogik, 1857. 
98 


THE PRESENT TEACHING OF ARITHMETIC 99 


by giving quite extended lists of divers virtues cul- 
tivated by the simple science of numbers. 

But it sometimes seems as if these discussions have 
been more harmful than beneficial When we hear 
some second year class dawdling along through a 
little simple number work, which no doubt has been 
elegantly developed, and out of which ethical and 
logical and general culture values have no’ doubt been 
duly extracted, we are forced to wonder whether in 
a maze of secondary purposes there is not lost the 
primary purpose—that of leading the child to “ figure” 
quickly and accurately in the common problems of 
his experience. 

The number concept— The fundamental principle in 
the method of teaching primary arithmetic has its 
root in the essence of number.t No one now 
affirms that number is an object of sense-perception,? 
although upon this inherited notion are based not a 
few of our present ideas as to method. ‘“-The notion 
of number is not the result of immediate sense-per- 
ception, but the product of reflection, of an activity of 
our minds. We cannot see nine. We can see nine 
horses, nine feet, nine dollars, etc., that is we see 
the horses, the feet, the dollars, if they are presented 
to us; that there are exactly nine, however, we cannot 


1 Beetz, K. O., Das Wesen der Zahl als Einheitsprinzip im Rechen- 
unterricht. Neue Bahnen, VI. Jahrg., 201. 
* McLellan and Dewey, p. 61. 


100 THE TEACHING OF ELEMENTARY MATHEMATICS 


see. If we wish to know this we are forced to count 
the things; and since we usually do this with the help 
of our eyes, the idea has got abroad that we see 
number.” ? 

In line with this idea we would be justified in say- 
ing that ove is not, primarily, a number, and it is 
historically interesting to know that only recently has 
it been so considered. The classical definition o# 
number is “a collection of units,’ ? a definition scien- 
tifically worthless. 

But while we put number into objects, on the other 
hand we derive our idea of number only from the 
presence of the world external to the mind. We see a 
group of people, and we begin by making an abstrac- 
tion (“ people”’), and we say, ‘“‘ Here are ten people” — 
thus calling them all by the one abstract name, even 
though the individuals be very different. “A careful 
observation shows us, however, that there are no 
objects exactly alike; but by a mental operation of 
which we are quite unconscious, although it holds 
within itself the entire secret of mathematical ab- 
straction, we take in objects which seem to be alike, 


1 Fitzga, E., Die natiirliche Methode des Rechen-Unterrichtes in der 
Volks- und Biirgerschule, I. Theil, Wien, 1898. This is one of the most 
common-sense books on method that has appeared in a long time. 

2 This is found in most of the older arithmetics. For example, Gemma 
Frisius, in his famous text-book, says, “ Numerum autores vocant multitu- 
dinem ex unitatibus conflatum. Itaque unitas ipsa numerus non erit.” 
Arithmeticae Practicae Methodus Facilis, Witebergae, M.D. LI, pars prima. 


THE PRESENT TEACHING OF ARITHMETIC IOI 


rejecting for the time being their differences. Here 
is to be found the source of calculation.”! So the 
idea of number is generated in the mind by the sense- 
perception of a group of things supposed to be alike.” 

Hence while we do not have a sense-perception of 
number, on the other hand few now attempt to teach 
number without the help of objects for the formation 
of groups. What these objects shall be is more of a 
dispute to-day than ever before. In Germany the use 
of numeral frames has been carried to an extent not 
known in America, and several forms of apparatus 
have been devised. But however valuable these aids 
may be in the first grade, it is doubtful if there is 
any excuse for their extensive use thereafter.2 In 
America the tendency has been along the Pestaloz- 
zian line, of taking any material that is at hand, 
although objection has been made to the most 
natural means of all, the fingers. Frequently, how- 


1 Laisant, La Mathématique, p. 15, 18, 19, 31. 

2“ Yede Zahl ist der Inbegriff einer gewissen Menge von Einheiten. 
Einheiten im Sinne des ersten Rechnens sind wirkliche Dinge.... 
Ein grundlegender Rechenunterricht ohne Veranschaulichung ist... 
undenkbar.” Grass, J., Die Veranschaulichung beim grundlegenden 
Rechnen, Miinchen, 1896, p. 5, 6. 

8 One of the best brief historical discussions of numeral frames is given 
in Grass, op. cit., 61 seq. The matter is discussed in Payne’s transl. of 
Compayré’s Lectures on Pedagogy, p. 384-385, the note on p. 385 being 
misleading, however. 

Die Finger sind das natiirlichste und niachste Versinnlichungs- 
mittel, Fitzga, I, p. 18. 


102 THE TEACHING OF ELEMENTARY MATHEMATICS 


ever, teachers have fallen into the error of forgetting 
Busse’s valuable suggestion, that the objects should 
not be such as to take the child’s attention from the 
central thought. At the same time, they should be 
such as relate to his daily life and such as have 
some interest for him.! 

There has also been a tendency in America to 
follow Grube to the extreme of using objects long 
after’ there is any need for -them. Some have 
devoted much energy to bringing children to recog- 
nize at a glance the number in a group, say nine, 
and this has connected itself with the best form of 
grouping to establish number relations and to enable 
the eye to grasp the group readily. A consideration 
of the forms 


patel 


shows how much more readily the eye grasps some 
forms than others. But after all, this is fundamen- 
tally the recognition of a familiar form, which we 
have learned has a certain number of spots, rather 
than the recognition of a number. In a game of 


1 Was durch das Leben in Schule und Haus und ausser dem Hause 
in den Erfahrungskreis des Kindes gekommen ist, auch das kann fiir das 
Rechnen verwertet werden. Alle Teile des Gedankenkreises sollen rech- 
nerisch durchleutet werden, in denen ihrer Natur nach Zahlen eine Rolle | 
spielen. Rein, Pickel and Scheller, Theorie und Praxis des Volksschul- » 
unterrichts, I, p. 361. 


THE PRESENT TEACHING OF ARITHMETIC 103 


cards we recognize the form of the nine as we do 
the form of the knave; we do not stop to count the 
spots, nor could we tell the number on a different 
arrangement unless we counted.! 
~ The uselessness of carrying this objective work too 
far is apparent when we consider that we never get 
our ideas of numbers of any size from thinking of 
groups; we get them from thinking of the relative 
places which they. occupy in the number series, or 
the time which it takes to reach that place in run- 
ning up that series, or the length of the line which 
would represent that number in comparison with unity.” 
Recently, sustained by high psychological authority, 
the effort has been made to make prominent the ratio 
idea from the very outset. That ratio is number is 
evident; that the converse is true, has the authority 
of Newton’s well-known definition; that a child should 
first consider number in this way has its advocates. 
“The fundamental thing,” says one of these “(in 
teaching arithmetic), is to induce judgments of rela- 
tive magnitudes.”? But such a scheme substitutes a 


1 Tf one cares to enter this field with any thoroughness, historically and 
psychologically, he should read Grass, op. cit., p. 14 seq., one of the best 
discussions available. 

2 Um uns gréssere Zahlen ohne Wiederholung des Zahlens etwas deut- 
licher zu vergegenwartigen, greifen wir daher zu dem Auskunftsmittel von 
Substitutionen. Das gebrauchlichste ist, fiir Zahlvorstellungen Zeitvor- 
stellungen zu substitutieren. Fitzga, I, p. 16. 

8 Speer, W. W., The New Arithmetic, Boston, 1896. 


104 THE TEACHING OF ELEMENTARY MATHEMATICS 


complex for a simple number idea, it is contrary to 
the historical sequence (whatever that may be worth), 
and it makes use of a notion of number entirely dif- 
ferent from that of which the child will be conscious 
in his daily life. It founds the idea of number upon 
measurement, but in so doing it uses the word measure 
in its narrowest sense. It makes use, also, of sets of 
objects (in the systems thus far suggested) by which 
is accomplished no more than Tillich accomplished 
with his blocks, while their character is such as to 
take the attention from the central thought of zumder. 

Fundamentally, as Laisant has pointed out, and 
Comte before him, the two notions of counting and 
measuring are the same.! The estimation of a mag- 
nitude directly by comparison is, however, extremely 
rare; “it is the zzdzrect measure of magnitudes which 
characterizes mathematics.” As to the necessity for 
the ratio idea at some time in the pupil’s course, 
there can be no question; the argument lies only as 
to where the idea should be brought in. The most 
temperate and philosophical discussion of the subject 
is that given by McLellan and Dewey in their “ Psy- 
chology of Number” (1895), a work which should be 
read and owned by every teacher in the elementary 
grades. It makes number depend upon measurement, 
but it uses this word in the broader sense indicated 


1 Laisant, La Mathématique, p. 17. 
2 A brief but very good discussion is given in Beetz, op. cit., p. 299. 


THE PRESENT TEACHING OF ARITHMETIC 105 


by Comte, including counting as a special form. In 
counting, however, it wages war against the “fixed 
unit’? system which the authors brand with Grube’s 
name, although Grube is by no means the father of 
it. It actually (as all do theoretically) substitutes the 
method of ¢himgs for the method of symbols, the 
Pestalozzian idea of numbers instead of figures, and 
it leads a general attack against the inherited weak- 
nesses of the traditional primary arithmetic. The 
work seems not to seek to place upon the child the 
burden of the ratio idea at the outset, but rather to 
lead him to a common-sense notion of number with- 
out fixed unit, of counting in the best form of the 
Knilling-Tanck school, of applying the knowledge of 
number to things instead of to relations of volumes 
and lengths. To count things; not to say 3+ 5 =?, 
but 3 cts. + 5 cts. = how many cents ?, or 3 five-cent 
pieces + 5 five-cent pieces are how many five-cent 
pieces?— this is to use number as the world first 
used it, to use number with a varying unit, to get an 
introduction to ratio at the best.1 Laisant sums up 
the matter of the proper place for the ratio idea 
when he says: “It is proper to ask if the idea of 
ratio, usually assigned place rather late in the study 
of arithmetic, does not deserve to be considered early 
in the course as a consequence of the notion of number.” * 


1 Fitzga, p. 28; McLellan and Dewey, p. 78, 147, 149, etc. 
2 La Mathématique, p. 30. 


1060 THE TEACHING OF ELEMENTARY MATHEMATICS 


When in elementary work we are led to feel that 
a child must not only think of a group of things or 
a ratio when he is learning about the numbers from 
I to 10, but that he must continue to think of groups 
and ratios, and to refer to objects, as he progresses, 
we impose upon him what no mathematician takes 
upon himself. The child must get his first notion of 
numbers from counting ¢kzzgs, as the world did; these 
things may in themselves be groups; in counting he 
really measures the group by the unit with which he 
is working; he gets a ratio, if we please to call it so, 
although the concept is not simple enough to be thrust 
upon him. But once the idea of number is there, it 
is then largely a matter of the number series; we have 
an idea of forty-seven as lying between forty-six and 
forty-eight, a little below fifty, and as being a number 
about half way (distance) to a hundred, and we have a 
vague idea that it would not take long to count it, 
about half as long (time) as to count a hundred. Thus 
we place it in a series, on a line, or in the flow of 
time, and thus we get an idea of its magnitude; but 
few people visualize it as a group of objects, and why 
should a child be asked to do so? 

Advocates of the idea that number means merely 
the how-many of a group, or the ratio of lengths 
merely, are disappearing as such scientific writers as 
Grassmann, Hankel, G. Cantor, and Weierstrass are 
coming to be known. The doctrine of “one-to-one 


THE PRESENT TEACHING OF ARITHMETIC 107 


correspondence” is being understood by elementary 
teachers, and it is not without suggestiveness in simple 
work in arithmetic. To the xwméer of a group cor- 
responds one xame and one symdol, as 


ee . 
e five 5 
e °@ 
If we establish the laws of these numbers, as that 


eo °® 2 e @ @ 
e and ©® equal @ and e 
® e @ @ @ (=) 


and give to a certain operation one name and one 


) 


symbol (as “addition,” +), then we may work with 
symbols according to these laws, and we need have 
no thought of the names or the numbers, but can 
translate back into numbers at any time we choose. 
Indeed, our symbols may force us to establish new 
kinds of numbers, as when we run up against the 
symbols 4—6, or \/4, or try to divide the circumfer- 
ence of a circle by the diameter. This notion of “one- 
to-one correspondence,” while not consciously one of 
elementary arithmetic, exists there just as really as it 
exists in later work. It does not take long for the 
child to “substitute for the reality of things the 
creatures of reason, born of his own mind.” In solv- 
ing a problem, be it one in the calculus, in algebra, 
or in the second year of arithmetic, we begin by sub- 
stituting for the actual things certain abstractions 
represented by symbols; we think in terms of these 


108 THE TEACHING OF ELEMENTARY MATHEMATICS 


abstractions, aided by symbols, and finally from our 
result we pass back to the concrete and say that we 
have solved the problem. It is all a matter of “one- 


) 


to-one correspondence,” it being easier for us to work 
with the abstract numbers and their corresponding 
figures than to work with the actual objects. Funda- 
mentally the process is something like this : 

1. By abstraction we pass to numbers. 

2. Thence we pass to symbols, and we make an 
equation, either openly, as in algebra, or concealed, 
as in many forms of arithmetic. This equation we 
solve, the result being a symbol. 

3. We find the number corresponding to this sym- 
bol, and say that the problem is solved. 

All this does not mean that primary number is to 
be merely a matter of symbols. It means that in 
mathematics we find it more convenient to work 
purely with symbols, translating back to the corre- 
sponding concrete form as may be desired. And so 
those teachers who fear lest the child shall drift into 
thinking in symbols instead of in number, are really 
fearing that the child shall drift into mathematics. 
In a rough way we may summarize the conclusions 
of the writers to whom reference has chiefly been 
made, as follows: 

1. Let the child learn to count things, thus getting 
the notion of number. These things are, for the pur- 


1 Laisant, La Mathématique, p. 20, 21. 


THE PRESENT TEACHING OF ARITHMETIC 109 


pose of counting, considered alike, and they may be 
single objects or groups. 

2. Let him acquire the number series, exercising 
with it beyond the circle of actually counted things. 

3. In the learning of symbols it does not seem to 
be a matter of moment as to whether these are given 
with the first presentation of number or not. They 
must, however, be acquired soon. 

4. Unconsciously and gradually the child will 
acquire the idea (never expressed to him in words) 
of the one-to-one correspondence of xzzsmber, name, 
symbol, and thereafter the pure concept of number 
will play a small part in his arithmetical calculations. 

5. The ratio idea of number should be introduced 
early, and applied in the work with fractions. 

The great question of method— M. Laisant has tersely 
expressed what is probably in the minds of most suc- 
cessful teachers of elementary mathematics, in the 
following words: ‘‘There are not, I believe, many 
methods of teaching, if by teaching we are to under- 
stand the exsemble of efforts by which we seek to 
furnish with accurate knowledge a human mind which 
has not yet reached its full degree of development. 

The problem is always the same :—to interest 
the pupil, to induce research, to continually give him 
the notion, the illusion if you please, that he is dis- 
covering for himself that which is being taught him.” ? 


1 La Mathématique, p. 188, 189. 


II0 THE TEACHING OF ELEMENTARY MATHEMATICS 


As for the rest, it is largely a matter of psycho- 
logical presentation and detailed device. Shall we 
extract square root by the diagram or by the formula? 
—The question is of relatively little importance in 
comparison with the great questions of method and 
of psychological presentation. So with most of the 
questions to be discussed in this chapter ; they are 
matters of detail which one teacher may work out 
one way, and another a different way, and the differ- 
ence in result may be so slight that the world has 
not been able, after centuries of experiment, to decide 
which is better. These matters vary with classes, 
with the advancement of pupils, and with the temper- 
ament of the teacher. To give simplicity of form with 
depth of thought is one of the qualities of the diffi- 


cult art of teaching, and it depends upon the individ- 


ual to attain to this simplicity.! 

The advance in the modern teaching of arithmetic 
is due much more to the recognition of the definite 
aim than to the discovery of improved methods. On 
the other hand, the influence of such writers as De 
Garmo and the McMurrys in America, opening up 


1“Tes moyens matériels, les procédés pédagogiques 4 mettre en 


ceuvre pour obtenir le résultat désiré sont €minemment variables, suivant — 


la nature des classes, 1’avancement des éléves, et aussi d’aprés la | 


maniére de voir et le tempérament du professeur. . . . Cette concilia- 
tion de la simplicité dans la forme avec la profondeur des idées con- 
stitue l’une des qualités de l’art difficile de l’enseignement.” Laisant, 


p. 192, 194. 


THE PRESENT TEACHING OF ARITHMETIC II! 


the German (and particularly the Herbartian) views 
of the bases of method, or the basis of education, 





has given a great impetus to teaching in general, 


and as a consequence has improved the teaching 
of arithmetic. For the application of these views to 
special lessons in number the reader is referred to 


the works of these writers.! 





The whole question of the formal steps to be taken 


by a teacher in presenting a new subject to a class 





should be considered apart from a work like this.? 
Suffice it to say here that Rein, whose presentation 
of the matter is as well known as any, sets forth five 
formal steps in the development of a lesson: 1. Prepa- 
Le ; 2. Presentation; 3. Association; 4. Condensa- 
tion ; 5. Application. Since the English translations 
Ihave given the application of the Herbart method to 
primary work only, the following translation of a fifth- 
grade lesson may be of value. 

_ Aim. How shall we write 12 tenths of a litre? 


| 1. Preparation. We can write 31,31, etc. Instead 


1 De Garmo, Chas., The Essentials of Method, p. 117; McMurry, C, A. 
a F, M., The Method of the Recitation, p. 19. For the best working 
out of the subject, however, one must consult Rein, Pickel and Scheller, 
‘Theorie und Praxis des Volksschulunterrichts, 6. Aufl., Leipzig, 1898. 
A brief statement of the application of the formal steps to elementary 
arithmetic is given in Briutigam’s Methodik des Rechen-Unterrichts, 





2. Aufl, Wien, 1895, p. 16, and in several other similar works. 

| ®The matter is clearly presented, historically and with comparative 
tables, in De Garmo’s Herbart, New York, 1896, Chap. V. 

: 


| 
| 


112 THE TEACHING OF ELEMENTARY MATHEMATICS 


of £1. we can also write 141; instead of $1, 1}1,! 


10 
5 
Also 4£1? | 


2. Presentation of the new. 412 or 1% can also be, 


etc. In what other way can we write 121.? (1,1)/ 


written another way. We already know that 4 can; 
be written 0.2. Further examples. What does a, 
figure before the decimal point indicate? One after 
the decimal point? | 

3. Association. Compare the way of writing 1451 
and 1.11; 38,1. and 3.31. Compare 1}1. and 1.2 lL 
Can we write 141. as we write 1,1.? 

4. Condensation. If we have to write more than! 
9g tenths of a litre we reduce the tenths of a litre to! 
whole litres, or to wholes and tenths, and we place: 
a decimal point between the wholes and the tenths: 
(or before the tenths, or after the wholes). A fourth 
or an eighth of a litre we cannot write as tenths, 
The figures after the dot always indicate tenths. 

6. Application. Read 0.4; 0.6. Read, as mixedi 
numbers, 2;3);°4:6.. Reduce to tenths 2:3; 476; Wtite’ 
24 wholes and 7 tenths. Write, as a mixed a. | 
22tenths ao Read,; as tenths.nt 2592:3)t age | 

The writing of numbers— Since Pestalozzi’s time’ 
there has been a controversy among teachers as to: 
whether a child should be taught the Hindu numerals: 
along with the numbers themselves.  Pestalozzi, as: 
we have seen, postponed this writing until the child’ 


1 Rein, Pickel and Scheller, Theorie und Praxis, V, p. 237. 


THE PRESENT TEACHING OF ARITHMETIC 113 


had a knowledge of the first decade. His argument, 
the limit sometimes being changed to five, meets with 
much approval among some of our best educators 
to-day. Many even go so far as to use the common 
symbols of operation and relation before the Hindu 
numerals are learned, giving forms like 


® e@ °@ 
peteuen wens PUTT — V0) = Ti 
Hh UT = UNM Lim TEENNY = U1 


Others ask, and with reason, why a symbol like 
x should be used, but not one like 4. Still others 
say, also with much reason, that the common psy- 
chological law of association is ample warrant for 
placing before the child, simultaneously, the forms 


Hl Four 4 


so he may see the “one-to-one. correspondence,” and 
fix the idea, the name, and the symbol together. 
This view is taken by Hentschel, one of the leading 
German writers upon method in arithmetic. “The 
pupils,’ he says, “have now seen the individual num- 
bers represented in three ways, and have so repre- 
sented them for themselves, namely, (1) by rows of 
marks, points, etc., (2) by number pictures, and (3) by 
figures. There now arises the question as to which 
of these three forms shall be used by the little ones 


in their first computations. Can we at once put 
I 


114. THE TEACHING OF ELEMENTARY MATHEMATICS 


them into work with the figures? For myself I an- 
swer, yes.”} 

The question, as is usually the case with these 
disputed matters of detail, is of relatively little im- 
portance. The experience of a century has left i@ 
entirely unsettled, the results being, so far as inves- 
tigations have shown as yet, quite as good in one: 
case as the other. It is easy to theorize upon such— 
a point, but it may be worth while to consider the 
difficulty which children have in connecting the num- 
ber itself with the proper symbol and especially with | 
the proper name in the number series, and hence to. 
‘make as much use as possible of the law of associa-' 
tion involved in presenting the number picture, the 
name, and the symbol simultaneously. 

The work of the first year—The majority of lead-: 
ing writers upon the subject limit the results of 
operations to the number space 1-10. Some go to: 
12. Others take the space I-20, and the argument | 
is a strong one that the foundation of all number: 
work lies in the mastery of the subject in this space. | 
Many advocate counting by tens during the second’ 
part of the year, and then filling in the series, thus 


1 Klotzsch, Hentschel’s Lehrbuch des Rechenunterrichts in Volks- ‘ 
schulen, 14. Aufl., Leipzig, 1891, p. Io. | 

2 Eg. Grass, J.. Die Veranschaulichung beim grundlegenden Rechnen, , 
Miinchen, 1896. This work gives a brief but valuable résumé of the 
leading theories of first grade work. | 


THE PRESENT TEACHING OF ARITHMETIC I1§ 


giving the child a number space beyond that in which 
he is actively working. Such a plan adds to the 
child’s interest, and allows him to teach himself by 
the talk of the home. On the whole, present experi- 
ence seems to show that the number space 1-20 for 
operations, with counting forward and backward in 
the space I-100 as recommended by Tanck, Knilling, 
and others, forms the limit of the working curriculum 
of the first year. Whether this limit can be reached 
depends entirely upon the class of pupils and the 
ability of the teacher. But to attempt to confine not 
only the results of operations, but also all ideas of 
number to the space 1-10, for the whole year, is 
not only unnecessary, but it is stupid and _ tedious 
for the children. 

The great desideratum in the first year’s work is 
facility in handling xumbders, not in solving applied 
problems. “Tell me a story about four,” is harmless 
enough at first, although there is no “story” told; 
but it gets to be a very old story before the year is 
done. Children like rapid work in pure number; one 
has but to step into a class whose teacher is awake 
to this idea, to realize the fact; and to dawdle through 
the year with nothing but “story” telling about num- 
ber not only leaves ungratified a natural desire, but it 
‘sows the seed of poor number work thereafter. There 
has nothing appeared in America for the last few years 
\that, considering its brevity, has done so much for the 


} 
/ 


| 


116 THE TEACHING OF ELEMENTARY MATHEMATICS 


better teaching of the subject as President Walker’s 
little monograph on “ Arithmetic in Primary and Gram- 
mar Schools.”! He cared little for theories and meth- 
ods, but he went to the root of the subject in a number 
of his observations. ‘“ At the present time the results 
in accuracy, if not in facility, of arithmetical work leave 
very much to be desired. Scarcely has the child been 
taught to count as high as ten, when he is put at 
technical applications of arithmetic, to money coins, to 
divisions of time, space, etc.; and these technical appli 
cations are increased in number and in difficulty through: 
the successive years of the grammar school, until for a 
large amount of so-called arithmetic the pupil gets com- , 
paratively little practice in the art of numbers.” ? This 
must not, of BASS be construed to mean that the child 
is to have no applied arithmetic; it is simply a protest 
against the neglect of that thorough drill in pure num- 
ber necessary to make a good calculator. ' 

The time for beginning the study of arithmetic is at 
present a matter of dispute. Should the first year of 
the subject, above mentioned, be synchronous with the. 
first school year? The “Committee of Fifteen” think 
not, and they recommend beginning with the second 
school year. Before Pestalozzi, as already said, the. 
subject was not begun until the child could read. Pes- | 
talozzi, however, recognized that the child has as much i 
taste for numbers as for letters, and proceeded to gratify | 


1 Boston, 1887, 2 POLL 





THE PRESENT TEACHING OF ARITHMETIC I17 


this taste in the first school year, a plan which has gen- 
erally been followed since his time. This idea of post- 
poning the formal study of number until the second 
year is one of several pre-Pestalozzian ideas which have 
recently appeared, and it has not as yet impressed 
itself upon educators as one of great importance. That 
the practical results for arithmetic, if the child con- 
tinues to the seventh grade, will probably be equally 
good, is true. That the child might put his twenty 
minutes a day, now devoted to arithmetic, to better 
use, may be true; but that he would do so is improb- 
able. Until we systematize play, and put the time 
gained from primary number to physical exercise, in 
the open air, under a skilled teacher, it is doubtful if 
the child should give up the few minutes a day in a 
line of work for which he has a taste and about which 
he delights to know. 

Oral arithmetic— The oral arithmetic, so necessary 
before the Hindu numerals made written computation 
easy, fell, as we have seen, into disfavor at the Renais- 
sance. Revived by Pestalozzi and his contemporaries, 
it had much favor not only in Europe, but also, thanks 
to Colburn’s excellent work, in America. But the 
advent of cheap slates and paper and pencils seems 
to have driven it out of our schools for a generation. 
It is now reviving, and it is to be hoped that we 
shall not again cease to secure reasonable facility in 
rapid oral work with the ordinary numbers of daily life. 


118 THE TEACHING OF ELEMENTARY MATHEMATICS 


The subject can easily be carried to an extreme; but 


within reasonable limits it should be demanded in every 


gerade. It lubricates the arithmetical machine, and five 


minutes a day to this subject could hardly fail to bring | 


all pupils to reasonable facility with numbers. 
Treating the processes simultaneously — This is, of 


course, as impossible as it is to have several bodies_ 


occupy the same space at the same time. But the 


expression means the so-called mastery of a number, 


the study of the four processes, before the next is 


studied. As already stated, this is the essence of the 


Grube method, its fundamental feature as well as its’ 


fundamental defect. “It seems absurd, or worse than 


absurd, to insist on thoroughness, on perfect number 
concepts, at a time when perfection is impossible .... 
If the child knows three, if he has even an intelligent 
working conception of three, he can proceed in a few 


lessons to the number ten, and will thus have all higher | 


numbers within comparatively easy reach.”? A more 


tedious way of presenting number than that of Grube’s | 


would be hard to find, and yet, in America and Ger- | 


many, this feature still has a considerable following. 


The spiral method—AIn the preparation of text- 


books we have had various experiments of late, all the | 


result of the restless desire to break away from the 


bad features of the older works. The so-called “spiral 


method”’ seems to have been first suggested by Ruh- , 


1 McLellan and Dewey, The Psychology of Number, p. 172, 176. 


THE PRESENT TEACHING OF ARITHMETIC 119 


sam,! and to have found little favor anywhere until it 
was recently taken up in America. It consists in 
taking the class around a circle, say with the topics of 
common fractions, decimal fractions, greatest common 
divisor, and square root; then swinging around again 
on a broader spiral, taking the same topics, but with 
more difficult problems; then again, and so on until 
the subjects are sufficiently mastered. 

The idea has much to recommend it. A child is 
not now expected to master common fractions by going 
once over the subject and then leaving it forever. And 
yet the older text-books expected him to do just that 
for greatest common divisor, square root, etc. But the 
idea can easily be carried to an extreme, the class 
swinging around the spirals so frequently as to pro- 
duce mathematical nausea. It is a question how 
elaborate the scheme should be made, and it has not 
been sufficiently tried to answer this question. 

Common vs. decimal fractions—The question of 
sequence of common and decimal fractions is one 
which has recently been much discussed. It is easy 
to dismiss the whole subject by some such remark as, 
“Logically the decimal fraction comes first, because it 
grows naturally out of our number system,” and this 
is frequently done in some educational sheets. Another 


1 Aufgaben fiir das praktischen Rechnen zum Gebrauch in den un- 
_ tern drei Klassen der Realschulen und in den obern Klassen von Biir- 


_ gerschulen in drei concentrisch sich erweiternden Cursen, 1866. 


120 THE TEACHING OF ELEMENTARY MATHEMATICS 


will say that the Prussian educational decree of 1872 
put the decimal fractions first, and that the experience 
of these many years has proved the wisdom of the 
plan. But just as strong an argument can be advanced 
by saying that psychologically the common fraction 
should precede, because the concept is the simpler; 
that historically it was in use long before the decimal 
system of writing numbers was known, to say nothing 
of the decimal fraction; and that Prussia’s experiment 
has been productive of such doubtful results that Baden, 
and Bavaria, and Saxony still follow the older plan.} 
The question is really, however, one belonging rather 
to the old-fashioned course than to the modern, to the 
days when the pupil was expected to “master” com- 
mon fractions before studying the decimal. Our 
modern arithmetics, of any standing, follow no such 
plan. The fact is, no one ever thinks, practically, of 
teaching 0.5 before 3, or 0.25 before +. The simple 
fractions 4, ¢, enter into the work of the first year; the 
forms 0.5, 0.25, represent a much greater degree of ab- 
straction, and hence should have place considerably later. 
But on the other hand, as between adding 0.5 and 
0.25, or 47% and 321, there can be no question as to 
which should have first place. And hence the con- 
clusion will probably be reached by most teachers that 


| 


1 For details as to these state systems see Dressler, Der mathe- 


matisch-naturwissenschaftliche Unterricht an deutschen (Volksschullehrer-) © 


Seminaren, Hoffmann’s Zeitschrift, XXIII. Jahrg., p. 15. 


THE PRESENT TEACHING OF ARITHMETIC I21 


the elementary treatment of simple fractions has the 
first place, but that, long before the pupil comes to the 
serious difficulties of the common fraction, the tables 
of United States money, or possibly those of the metric 
system, should make him familiar with the decimal 
forms and the simple operations therewith. 
Improvements in algorism, that is, in the arrangement 
of work in performing the elementary operations, are 
constantly appearing, and some are of real value. Two 
which are now struggling for acceptance, with every 
prospect of success, may be mentioned here as types. 
In subtracting 297 from 546, we have the 


546 
two old plans, both dating from the time of 207 
the earliest printed text-books, at least. The Be 


calculation is substantially this: 

f7-trom 16,°9; 9 from 13, 4; 2: from: 4,°2% ‘or 

eeezirom 16, 90; 10 from 14, 4;°3 from 5, 2. 

But we have also a more recent plan: 

weezeand 0, 163.10 and 4; 14; 3-and 2, 5. 

To this might be added a fourth plan which has 
some advocates: 

4. 7 from 10, 3; 3 and 6,9; 9 from Io, I; I and 3, 
Gee trom 4, 2. 

All four of these plans are easily explained, the 
first rather more easily than the others. But the 
third has the great advantage of using only the addi- 
tion table in both addition and subtraction, and of 
saving much time in the operation. It is the so- 


122 THE TEACHING OF ELEMENTARY MATHEMATICS 


called “ Austrian method” of subtraction. The fourth 


plan, while a very old one and possessed of some. 


good features, is so ill adapted to practical work as 


to have no place in the school. It is hardly neces- 


sary to say that the old expressions, “borrow”’ and 


l 


“carry, in subtraction and addition are rapidly going 


out of use; they were necessary in the old days of 
arbitrary rules, but they have no advocates of any 
prominence to-day. 


In division we have also an “ Austrian method,” a 


valuable arrangement. It is not long since a prob- 
lem like 6.275 + 2.5 was “worked” by a rule which 


was rarely developed. Now the work is arranged in 


this way: 
2.51 
2.5)6.275 25)62.75 
50 
12.75 
L255 








0.25 
0.25 


oe 


Such an arrangement leaves no trouble with the 


decimal point, and the work is easily explained. In 


the above problem the entire remainder is brought 
down, and the decimal point is preserved throughout, 
as should be done until the process is thoroughly 
understood ; then the abridgment should appear. 


THE PRESENT TEACHING OF ARITHMETIC 123 


The explanations of greatest common divisor, divi- 
sion of fractions, etc., are so fully given in any of 
our recent American text-books that it is not worth 
while to attempt them in a work of this nature. | 

The formal solution of applied problems is now 
generally recognized as logic work as well as number 
work. The result of the problem is as important as 
ever, but it is not all-important; the value of a logi- 
cal explanation is now recognized—of course when 
the pupil has reached the proper grade. Hence the 
solutions of problems in percentage and in analysis 
are now generally given in step form, the actual 
work of the elementary operations being omitted. 
For example: 

A commission merchant remits $1073.50 as the net 
proceeds of a sale after deducting 5% commission; 
required the amount received from the sale. 


I. 0.95 of the amount = $1073.50. 

2 ee the amount = $1073.50 + 0.95 = $1130, 
by dividing these equals by 0.95. 
' Or better still, by letting + represent this amount 


(not the xwmber of dollars, since we are preserving 
the dollar sign before the other numbers), 


I. 0.95% = $1073.50 
2) es b1073.50 + 0.95 
= $1130. 


124 THE TEACHING OF ELEMENTARY MATHEMATICS 


This introduces the equation form in a more pro- 
nounced way, but this is now generally approved by 
educators.} | 

There are still some advocates of the following 


plan: 

iif 95% of the amount is $1073.50. 
2... 1% of the amount is x of $1073.50 = $11.30. 
3. .. 100% of the amount is 100 x $11.30 = $1130. 


This, the unitary method, is by some thought to 
be simpler than the others, though why it is simpler 
to derive 0.01 from 0.95 than to derive 1 from 0.95, 
it is difficult to say. 

The following form has also an occasional advo- | 


cate: 
I. Let 100% equal the amount. | 
2. Then 100% —5% = 95%. 
3. If 95% = $1073.50, 
4. Ly = pe 1 1.30, 


5. and 100% = $1130. 


This is a relic of the medizval method of “false posi- 


) 


tion,” a pre-algebraic device. The 100% is merely 1, , 


and we begin by letting this 1 equal the unknown: 


1“ Alle Padagogen sind hierin einverstanden.” Hentschel, p. 81. 
“Can any one imagine a good teacher, who is also a good algebraist, who 
will not train his pupils to use letters for numbers long before arithmetic , 
is completed?” Safford, T. H., Mathematical Teaching, Boston, 1837, 
p. 23. The question is discussed in a broad way by Schuster, M., Die, 
Gleichung in der Schule, in Hoffmann’s Zeitschrift, XXIX. Jahrg., p. 81. 


THE PRESENT TEACHING OF ARITHMETIC 125 


quantity. Of course x or any other symbol might be 
used to better advantage, for we know very well that 
the unknown quantity is wot 1. Furthermore, 95% 
does not equal $1073.50; it is 95% of the amount, 
or of x, that equals $1073.50. 

By following such a plan as the one first mentioned 
the well-founded complaint against the thoughtless 
mechanism of the past disappears. Instead of words 
and rules without content, there is content with a 
minimum of words and with no unexplained rule.! 

It is only a few years back that such forms as 
Steere te fee Orsae tt oe sO Ito acu 
+8 sq. ft. =3 ft,” and the like were not uncommon. 
er. however, all careful teachers are insisting that 
such inaccuracies of statement beget inaccuracy of 
‘thought and hence should not be tolerated in the 
schoolroom. It is true that these all depend upon 
the definitions assumed, and that well-known teachers 
-have advocated such a change of definition as will 
allow of saying “4 ft. x 2 yds. = 3456 sq. in.’’?; but, 


_ 1Die Kinder... lésen einschlagige Aufgaben, aber alles das geschieht 
/meistens auf mechanischem Wege. Wir finden Worte und Regeln ohne 
Inhalt. Fitzga, p. 5. The other side of the case, the danger of using 
algebra unnecessarily is presented in Supt. Greenwood’s Dissent from 
Dr. Harris’s Report of the Committee of Fifteen. 

2 This illustration, from an article by Professor A. Lodge in the General 


Report of the Association for the Improvement of Geometrical Teaching, 





“January, 1888. Similar articles have appeared in Hoffmann’s Zeitschrift 
‘in recent years. 


| 
1 
; 


i 


126 THE TEACHING OF ELEMENTARY MATHEMATICS 


with our present definitions, such forms lead to great 
looseness of thought. 

It is the loose manner of writing out solutions, tol- 
erated by many teachers, that gives rise to half the 
mistakes in reasoning which vitiate pupils’ work. The 
carelessness in form begets that carelessness of thought 
which gives point to such amusing absurdities as these: 

1. A bottle } full =a bottle $ empty. Divide by 4, 

.. a bottle full = a bottle empty. 

2. 20 dimes = 2 dollars. Square each member and 

.. 400 dimes = 4 dollars.! 

Longitude and time furnish a type of the applied 
problems of arithmetic, one in which much careless- 
ness of form and thought is often apparent, and as, 
such it is entitled to some special consideration. | 

The subject is best presented, perhaps, by a brief 
discussion of the question of the relative positions of | 
the sun and earth at the hour of the class recitation, 
the globe being held before the class, the northern’ 
hemisphere visible, and North America being on the, 
lower half so as to be recognized easily (it being: 
then “right side up” to the pupils). The sun being) 
located, the question of the forenoon and the after-| 
noon on the earth’s surface may be discussed, then 
the position of midnight, then the effect of the revo-; 
lution of the earth with respect to these periods ; and | 


1 Adapted from Rébiére, A., Mathématiques et mathématiciens, 2. éd., 
Paris, 1893, p. 331. i 


THE PRESENT TEACHING OF ARITHMETIC 127 


finally, for one lesson, the number of degrees through 
which the schoolhouse and vicinity must pass in order 
that the time shall be 24 hours later. 

All this leads to the development of two tables, 
the foundations upon which the subject rests: 


TABLE I 
360° correspond to 24 hrs. 


. 1° corresponds to 345 of 24 hrs. = =, hr. = 4 min. 


Te cer 


‘. 1’ corresponds to 75 of 4 min. = = min.= 4 sec. 


pl 


-. 1’ corresponds to gy of 4 sec. = +g sec. 


TABLE IL 
24 hrs. correspond to 360°. 
%. I hr. corresponds to gy of 360° = 15°. 
“. I min. corresponds to gy of 15° =4 of 1°=15/. 
Bw sec. corresponds to g, of 15! =f of 1 = 15". 


: To say that 360° = 24 hrs. is as inaccurate as to 
say that $4=24 lbs. of beef; there may be some 
correspondence, aS. it uvalies etc.) DUL there; is) no 
such equality as is set forth in the statement. 

| The theory of the subject is now best brought out 
‘by numerous simple oral problems of this nature: If 
ithe difference in longitude between two ships is 10°, 
what is their difference in time? If their difference 
jn time is 20 min., what is their difference in longi- 
tude? To make such problems practical, cases of 


128 THE TEACHING OF ELEMENTARY MATHEMATICS 


ships or observatories should be used, since the recent | 
rapid development of standard time has shut out local 
time in the large majority of places in the civilized world. | 

Written solutions may now be required in some 


; 


such form as the following: 


The difference in longitude between two ships is” 
10° 45’ 30’, required the difference in time. 


Teo Sc Auemin <== 40 «min. | 
2. 45 X 7s min. = 3 min. (or 45 x 4 sec. = 180 SEC. 
= 3 min. ). 
3. 30 X qk sec. = 2 sec. 4. .. 43 min. 2 seq, 


The difference in time between two ships is 43’ 
min. 2 sec., required the difference in longitude. | 


1. 43 Xf of 1° = 103° = 10° 45! (or 43 x 15! = +0 ; 
Die 2th Oe a0 ie TOnrd Sako. | 
Some of the older arithmetics still write ‘‘2 hr,, 
3! xxl? for 2 hr, 3.min: 15 sec.,;or 2h, 3.1m, i 5aamp 
but it is unwise to change the general custom of, 
using the ' and " for longitude only. More serious, 
is their adherence to the mechanical rule, and to, 


such forms as these: ; 
43 min. 2 sec. 


LS OA 5 eee 30 15 
2hr. 3 min. 2 sec. 645! 30! 
= 43 min. 2 sec. = 10° 45/ 30! 


Explain all we will, such forms tell the eye that 
degrees divided by an abstract number give hours, 


THE PRESENT TEACHING OF ARITHMETIC 129 


and that time is transformed by some miracle into 
longitude by multiplying by 15! Text-book makers 
may argue for brevity, but the astronomer and the 
navigator who wish brevity always use longitude 
tables. It is not brevity that we seek; it is an 
understanding of the process. 

The two points at which the teacher needs to aim, 
after the elementary correspondence between _longi- 
tude and time is fixed, are (1) standard time, and 
(2) the date line. The old-style complicated prob- 
lems may well give way to these new and interesting 
topics. The last decade of the nineteenth century 
has seen standard time made well-nigh universal in the 
highly civilized portions of the world, and the recent 
events in the Philippines have given to the subject of the 
date line even greater interest for American pupils. 

Ratio and proportion still maintain their conven- 
tional copartnership in most of our arithmetics, 
‘usually setting forth an array of problems inherited 
from some generations past. There is just now a 





good deal said about introducing the ratio concept 





earlier in the course, and this may happily break up 

the partnership and show ratio as the important sub- 

ject which it really is. 

_ At present, in the standard type of arithmetic, 
1 For a full discussion of these two subjects, with late information con- 


4 cerning standard time, and with maps showing the date line, the reader is 
| referred to Beman and Smith’s Higher Arithmetic, Boston, 1897. 
/ K 


| 
i 


130 THE TEACHING OF ELEMENTARY MATHEMATICS 


ratio has place merely as an introduction to pro- 
portion. The latter subject is taught as a matter 
of rule, as if it were to be used so often as t@ 
justify this unscientific treatment. The fact is, the 
subject is rarely used in business, and almost its 
only arithmetical applications of value are to be 
found in physical problems and in problems involving 
similar figures. Before simple equations were invented 


the subject had much more value than at present, | 


and the arbitrary ‘Rule of Three,” as it was called, | 


may have been justifiable. At present, to teach the ' 


subject by mere rule, or by any such senseless device 


as the “cause and effect”? method, is unwarranted. 


There is just now a growing reform in presenting | 
proportion. This movement employs the fractional | 


notation, with which the pupil is familiar, and the 


common equation form, thus: 7% = ae to finds Mul 
3 e 


tiplying these equals by 3, += #4. 


Consider, for example, a single applied problem: , 


If a plumb line 1 yd. long casts a shadow 6 ft. long, 
how high is an adjacent flagstaff which at the same 
instant casts a shadow 84 ft. long? 


1. Let x =the xumber of feet required. 


Then ae or =n the ratio of the heights, 


84 ft. ne 84 


d 
ei. ETO Ore 


| a P= 3 os 


= the ratio of the shadow lengths. | 





THE PRESENT TEACHING OF ARITHMETIC 131 


2. And since the heights are proportional to the 
shadow lengths, 


WIR 
[| 
al 


3. Multiplying by 3, += 42. 
Berthe. stati is 42 it. high: 


After the class is familiar with the theory, the work 
should be given with the other symbols, because 
these are needed in common scientific reading, thus: 
4:3 = 84:6, or even the antiquated form +:3::84: 6. 

Solutions of this nature, with the reasoning set 
forth, give us the “ thought reckoning” (Denkrechnen) 
which our best educators demand, in place of the rule- 
work of the old school.t | 

Square root was formerly treated geometrically, 
that being the plan inherited from the Greeks, the 
nation which most excelled in geometry in ancient 
times.2. But the method which follows the algebraic 
formula is preferable on many accounts. The fact that 
‘the square on f+ is f2+ 2 fu +n*, where / stands 
for the found part of the root and # for the next 
figure, may profitably be pictured by a geometric 


1The general question of proportion is discussed in a valuable 

| article by Dressler, Der mathematisch-naturwissenschaftliche Unterricht 

an deutschen (Volksschullehrer-) Seminaren, Hoffmann’s Zeitschrift, 
XXIII. Jahrg., 1. 

2Theon of Alexandria, father of Hypatia, gave the common geomet- 

ric plan. Gow, History of Greek Mathematics, p. 55; Cantor, I, p. 460. 


132 -THE TEACHING OF ELEMENTARY MATHEMATICS 


diagram. But the formula is to be preferred to the | 


diagram, as a basis for work, because 


1. The geometric notion limits the idea of involution 


to the square and cube roots; 
2. The formula method makes the cube and higher 
roots very simple after square root is understood; 


3. We are working with numbers, not with geometric . 


concepts ; 


4. The formula lends itself more easily to a clear. 


explanation of the process. 


One of the great difficulties in explaining square root . 


lies in the fact that tradition has encumbered it with 
superfluous difficulties. Consider, for instance, the 
question, ““Why do we separate into periods of two 
figures each, beginning at the right?” The answer 
might be given, ‘We need not do so; it was neces- 
sary when square root was merely a matter of rule; 


if one thinks, such separation is quite unnecessary; - 


furthermore, we would not begin at the right anyway, 


but rather at the decimal point, this rule having been - 


framed long before the decimal point was known.” 


Again, ““Why do we bring down only one period at a. 
time?” For reply we may say, “We don’t; it is much _ 
better for beginners to bring down all of the remainder | 
each time, because it makes the explanation easier.” | 


Of course, after the complete process is fully under- 
stood we may adopt this and other abridgments if 


we desire, and then the explanation is not difficult; j 


THE PRESENT TEACHING OF ARITHMETIC ~ 133 


but it is very poor policy to let such unnecessary 
questions enter at a time when the teacher is seeking 
to have the process clearly understood. 

It may be said that these suggestions and the follow- 
ing solution make the process longer than necessary. 
But since almost the sole justification for the subject 
of involution is the fact that it offers training in logic, 
this training is of paramount importance. For practical 
purposes the square root is usually extracted by the 
help of tables. 

A problem in square root might, then, be arranged 
as follows: 


23.4 =root 
547.56 contains some square, f7+2 fu+n 
f? = 400 





2f=40 147.56contains 2 fz + n*, where f= 20 
2f+n=43 1290 =2fn+n" 
- 2f=46  18.56contains 2 fz+x*, where f= 23 
2ftn=46.4 18.56=2fn+n 


_ This arrangement shows what each number equals 
(exactly or approximately), and the only things to 





‘explain are (1) these equalities, and (2) why 2/ is 
‘taken as the “trial divisor,’ matters offering no diff- 
‘culties. 


\ 1F¥or full explanation, and for other suggestions as to the factoring 
method, treatment of fractions, the double sign, etc., see Beman and 
Smith’s Higher Arithmetic, Boston, 1897, p. 35. 


| 





134 THE TEACHING OF ELEMENTARY MATHEMATICS 


The metric system — The common measures of daily 
life demand great attention in arithmetic. Until they 
have become thoroughly familiar, until they have taken 
prominent place in the child’s mind, until they have 
been taught with the actual measures (as far as may 
be) in hand, and until they have been practically used 
in hundreds of concrete problems, the metric system 
has no place. The child can get along for a while 
without this system; indeed, he may never be con- 
scious of a loss if he does not know it; but the com- 
mon system he needs daily. 

On the other hand, as compared with the apothecaries’ 
and troy measures, or with leagues, furlongs, barley- 
corns, pipes, tuns, quintals, etc., the metric system 
should certainly have precedence. 

Only two or three bits of advice to the teacher need 
be given. First, these measures, like all others taught 
to the child, should be actually in hand; they must be 
made to seem real by abundant use; merely to learn 
the tables is of little value. The French schools, with 
their little cases of metric units on the front wall of the 
recitation rooms, always within sight of the children, 
set an example worthy of our attention.} 

Again, the child will probably use the system by 
itself if at all; that is, he will not be translating back 
and forth with the common system. To ask how many 
grammes in 4 cwt. 37 lbs. 2 0z., is worthless as a practical 


1See also Fitzga, I, p. 41, 57. 


THE PRESENT TEACHING OF ARITHMETIC 135 


problem; it gives the child a little “figuring,” but it 
destroys his appreciation of the great advantages of 
the modern system. A few of the common units may 
be translated, as in a question like this: A traveller in 
Germany is allowed 25 kilos of baggage free; about 
how many pounds is this? But such translation should 
be confined to common cases and to oral work. 

The pupil should be led to see that the names are 
not so strange as might at first appear. As a gas- 
metre measures gas, and a water-metre measures water, 
so a metre is a unit of measure ; it is a little longer than 
our yard. And 


as a mill is 0.001 of $1, so a millimetre is 0.001 of 
PaiuCiic 

as a cent is 0.01 of $1, so a centimetre is 0.01 of 
T.1netre; 

as a decimal point comes before tenths, so a deci- 
metre is 0.1 of I metre; 

as a dekagon is a 10-angled figure, so a dekametre 
is IO metres. 


So mzlli- means 0.001, deci- means O.I, 
centi- means O.OI, deka- means 10, 


and there are only three new prefixes to learn: 
| 


hekto-, which means 100, 
kilo-, which means 1000, 
myria-, which means 10,000. 


136 THE TEACHING OF ELEMENTARY MATHEMATICS 


With these prefixes well in mind the tables of the 


metric system are practically known. Hence a great 
deal of the oral drill in this work may profitably be 
devoted to these prefixes, taking them at random 
and asking their numerical equivalents, and vice 
versa. . 

The grade in which the metric system is taught is 
determined largely by the science work in the school. 
Since all science now uses this system, it may be taken 
up as soon as simple physical problems are introduced. 
But reference is so frequently made to the system in 
the current literature of the day, that to postpone the 
subject beyond the eighth grade, or to teach it in a 
perfunctory manner, is unwarranted. 

The applied problems, and especially the business 
problems involving percentage, are so well adjusted 
to the uses and capacities of the various grades, 


in the modern American text-books, that little need 


be said upon the subject. But topics like true dis- 
count, equation of payments, partnership, involving 
‘time, arbitrated exchange, insurance as it was fifty 
years ago—these subjects have no place in the com- 
mon school arithmetic of to-day. Our recent books 
generally print pictures of drafts, checks, notes, etc., 
and give such explanations of common business cus- 
toms as render these intelligible to pupils before they 
leave the eighth grade. Such helps, and the study 
of the actual documents in the classroom, will si- 


a Ee ed 


] 


THE PRESENT TEACHING OF ARITHMETIC 137 


lence much of the prevalent criticism that we teach 
too much for the school and too little for life. 

*¢Short cuts’?—The short methods so much sought 
in earlier times are now less in demand. The reason 
is not that time is considered less precious, but that 
the “short cuts’”’ have been found generally to apply 
to problems of no importance, or that the elaborate 
use of tables has rendered them unnecessary. For 
example, it was once considered a mark of an ex- 
pert accountant to have at hand numerous short 
methods of reckoning interest; now the accountant 
turns at once to his interest tables, and the average 
man with no tables at hand has forgotten the rules 
of his school days. 

Formerly the expression “75° + 15 = 5 hrs.” was 
allowed on the score that its brevity justified its 
falsity ; now, any one who has occasion to solve prob- 
lems of this kind in a practical way resorts to tables. 
Formerly, mere rule work was justified in square and 
cube root on the plea of brevity; now, for practical 
purposes, we generally extract such roots by loga- 
rithmic or evolution tables. 

Mensuration was formerly taught solely by rule. 
Even now the strictly scientific treatment belongs to 
geometry. But there are certain propositions that are 
so commonly needed that they must have place in 
arithmetic for those who may not study geometry. 


1 Vielfach nur fiir die Schule und nicht fiir das Leben. Fitzga, I, p. 6. 


138 THE TEACHING OF ELEMENTARY MATHEMATICS 


Such are the propositions which give the formulae, 
for measuring the square, or more generally the rec-. 
tangle and the parallelogram, the triangle, possibly the , 
trapezoid, the circle, the parallelepiped, the cylinder, 
and possibly also the cone and sphere. 

The mensuration of these figures may easily be. 
taken up in arithmetic in a reasonably scientific way, - 
and this is outlined in most of our modern text- 
books. For example, the computation of the area. 
of a rectangle 2 in. by 3 in. is easily made a matter 
of reason by using a figure illustrating the statement 
2 EXi3tx vrsdstin, ==.0\sdaiin}  orithe statements2a mms 
sq. In. = 6 sq. in. A parallelogram cut from paper 
is easily shown by the use of the scissors to equal 
in area the rectangle of the same base and same 
altitude, a figure already considered. By paper-cut- 
ting the triangle is shown to be equal to half of a 
certain parallelogram, and hence to half of the rec- 
tangle having the same base and the same altitude. 
By a few measurements of circumferences and their 
corresponding diameters the ratio ¢: @ can be shown 
to be approximately 34, a value sufficiently exact for 
ordinary mensuration. The teacher may then state, 
if thought best, that it is proved in geometry that a 
closer approximation is 3.1416, or 3.14159. The pupil 
has thus the interest of a partial discovery, and at 
the same time the possibilities of the more advanced 
mathematics are suggested. Similarly, as set forth in 


THE PRESENT TEACHING OF ARITHMETIC 139 


many of our better class of text-books, the other 
necessary propositions in mensuration may profitably 
be treated.t 

Text-books — In the days when text-books were few 
and poor there was some excuse for dictating elab- 
orate notes. The arithmetic copy-book was then an 
institution of some importance. But at present there 
is no such excuse; we have good books, and they 
save the time of pupil and teacher. This does not 
mean that the book shall be a master to be feared, 
but rather a servant to assist. In the lower grades, 
while the teacher should seek to follow the general 
lines of the text-book, each new demonstration should 
be discovered by the class (of course with the teacher’s 
leading) in advance of the assignment of book work. 
If the author’s plan is reasonably satisfactory it should 
be followed, in order that the pupil may be able to 
review the discussion without the waste of time in 
note-taking; a great many hours are squandered by 
teachers in attempting to “‘develop”’ something along 
some line not followed by the text-book in hand, 
when the author’s method is quite as good — usually 
better. There are now several excellent text-books 
with satisfactory demonstrations and with up-to-date 
problems, and these should receive the support of the 
profession. 


1 See also Hanus, P. H., Geometry in the Grammar School, Boston, 
1893. 


140 THE TEACHING OF ELEMENTARY MATHEMATICS 


But with any text-book we shall do well to keep 
in mind the words of President Hall: ‘ American | 
teachers seem to me to have spun the simple and 
immediate relations and properties of numbers over 
with pedantic difficulties. The four rules, fractions, 
factoring, decimals, proportion, per cent., and roots, 
is not this all that is essential? The best European 
text-books I know do only this, and are in the 
smaller compass, for they look only at facility in 
pure number relations, which is hindered by the irrele- 
vant material which publishers and bad teachers use 
as padding.” } 

Explanations — The question of the explanations to 
be given to and demanded from a child is a serious 
one. The primary work is preéminently that of lead- 
ing the child to discover the relations of number, and 
to memorize certain facts (like the multiplication table) 
which he will subsequently need. A few rules of 
action suggested by M. Laisant are worthy of atten- 
tion: “Follow a rigorously experimental method and 
do not depart from it; leave the child in the pres- 
ence of concrete realities which he sees and handles 
to make his own abstractions; never attempt to 
demonstrate anything to him;? merely furnish to him 
such explanations as he is himself led to ask; and 


1 Letter from G. Stanley Hall to F. A. Walker, in the latter’s monograph 
on arithmetic, p. 23. 
2 J.e., by a formal, logical demonstration. 


THE PRESENT TEACHING OF ARITHMETIC I4!I 


finally, give and preserve to this teaching an appear- 
ance of pleasure rather than of a task which is im- 
posed. If cerebral fatigue is produced, if the child 
is led to fix his attention on matters of no interest, 
and to master a line of reasoning too much in ad- 
vance for him, then the result is a failure.’ } 

The period of explanation comes later in the course, 
say after the fifth grade; but even here the explana- 
tion should rather be by questioning on the part of 
the teacher than by a full and free demonstration by 
the pupil. Where complete “explanations” are re- 
quired from the pupil, say of subjects like greatest 
common divisor, the division of fractions, cube root, 
etc., the result is usually a lot of memoriter work of 
no more value than the repetition of a string of rules. 
But by questioning as to the “why” of the various 
steps, the reasoning (which in most such work is all 
that is essential) is laid bare. 

It is the same with many applied problems. The 
set forms of analysis sometimes required of pupils 
is of very questionable value. On the other hand, a 
statement of the pupil’s own reasoning is, of course, 
extremely important, when he is sufficiently advanced 
to give it. But for primary children any elaborate 
explanation is impossible. Indeed, in the midst of all 
our theorizing on the subject of explanations, it is 
refreshing to read what a psychologist like Professor 


1 La Mathématique, p. 203, 204. 


142 THE TEACHING OF ELEMENTARY MATHEMATICS 


James has to say upon the subject of primary work:, 
“It is ... in the association of concretes that the, 
child’s mind takes most delight. Working out results 
by rule of thumb, learning to name things when they | 
see them, drawing maps, learning languages, seem to 
me the most appropriate activities for children under 
thintecn wie sbecensared: int su jae. elec! pretty confi- 
dent that no man will be the worse analyst or reasoner 
or mathematician at twenty for lying fallow in these 
respects during his entire childhood.” 4 

Approximations— There is a feeling among many 
teachers that some virtue attaches to the carrying of 
aresult to a large number of decimal places, and 
hence this is rather encouraged among pupils. As 
a matter of fact the contrary is usually the case in 
practice. If the diameter of a circle has been meas- 
ured correctly to’ 0.001 inch’ there is no use “ifm! 
attempting to compute the circumference to more 
than three decimal places, and 3.1416 is a better mul-' 
tiplier than 3.14159. The result should be cut off 
at thousandths and the labor of extending it beyond . 
that place should be saved. 

Now since we rarely use decimals beyond o0.oo1 
except in scientific work, and since zo result can be 
more exact than the data, and since even our scientific 
measurements rarely give us data beyond three or four 
decimal places, the practical operations are the contracted 


1 Letter to F. A. Walker, in the latter’s monograph, p. 22. 


THE PRESENT TEACHING OF ARITHMETIC 143 


ones, those which are correct to a given number of 
places. For this reason, in this age of science, ap- 
proximate methods are of great value in the higher 
grades which precede the study of physics. The fol- 
lowing are types of such work:! 





10.48 10.48 
3.1416 3.1416)32.92 = 31416)329200 
31.44 3142 
1.048 150 
0.419 126 
0.010 24 
0.006 24 
82,02 fais 


For the same reason the practical use of a small 
logarithmic table is of great value in the computa- 
tions of elementary physics. Two or three lessons 
suffice to explain the use of the tables and to justify 
the laws of operation, a small working table can be 
bought for five cents, and the field of physics affords 
abundant practice. 

Reviews— However much reviews may fail from 
their stupidity, as is apt to be the case with “set 
reviews,” a skilful teacher is always reviewing in 
connection with the advance work. But there is one 


season when a review is essential, a brisk running 


1 The explanations are given in any higher arithmetic, eg. Beman 
and Smith, p. 8, 11. 


144 THE TEACHING OF ELEMENTARY MATHEMATICS 


over of the preceding work that the pupil may take 
his bearings, and this is at the opening of the school 
year. Such a refreshening of the mind, such a lubri- 
cating of the mental machinery, gets one ready for 
the year’s work. Complaints which teachers generally 
make of poor work in the preceding grade are not 
unfrequently due to the one complaining; the effects 


———r—r 


of the long vacation have been forgotten; the engine _ 


is rusty and it needs oiling before the serious start 


is made. 


In these reviews the same correctness of statement 


is necessary as in the original presentation, though | 


not always the same completeness. To let a child 


say that 2+ 3 x2 is 10 (instead of 8) is to sow tares | 


which will grow up and choke the good wheat. To | 


let him see forms like 


2 taxi Sie ==.6 squit As) beg ree 
V4 sq. ft. = 2 ft., 2 x 0.50= $1, etc, 


or to let him hear expressions like ‘As many times 


as 2 is contained in $10,” “‘2 times greater than $3,” 


etc., is to take away a large part of the value that 


mathematics should possess. 


CHAPTER VI 
THE GROWTH OF ALGEBRA 


Egyptian algebra— Reserving for the following 
Arapter the question of the definition of algebra, we 
“may say that the science is by no means a new one. 
Or rather, to be more precise, the idea of the equa- 
tion is not new, for this is only a part of the rather 
undefined discipline which we call algebra. In the 
oldest of extant deciphered mathematical manuscripts, 
the Ahmes papyrus to which reference has already 
been made, the simple equation appears. It is true 
that neither symbols nor terms familiar in our day 
are used, but in the so-called “aw computation the 
linear equation with one unknown quantity is solved. 
Symbols for addition, subtraction, equality, and the 
unknown quantity are used. The following is an 
example of the simpler problems which Ahmes gives, 
his twenty-fourth: “ Haz (literally eap), its seventh, 
its whole, it makes 19,’ which put in modern sym- 


bols means 7 tee 19. Somewhat more difficult 


problems are also given, like the following (his 
thirty-first): “Hau, its 2, its 4, its 4, its whole, it 


makes 33,” 
te, Ze that+tr+r= 33. 


L 145 


146 THE TEACHING OF ELEMENTARY MATHEMATICS 


It must be said, however, that Ahmes had no 
notion of solving the equation by any of our present 
algebraic methods. His was rather a “rule of false 
position,” as it was called in medizeval times, — guess- 
ing at an answer, finding the error, and then modi- 
fying the guess accordingly... Ahmes also gives 
some work in arithmetical series and one example if, 
geometric. 

Greek algebra— Algebra made no further progress, 
so far as now known, among the Egyptians. But in 
the declining generations of Greece, long after the 
“golden age’ had passed, it assumed some impor- 
tance. As already stated, the Greek mind had a 
leaning toward form, and so it worked out a wonder- 
ful system of geometry and warped its other mathe- 
matics accordingly. The fact that the sum of the | 
first #2 odd numbers is 2%, for example, was dis- 
covered or proved by a geometric figure; square root , 
was extracted with reference to a geometric diagram; 
figurate numbers tell by their, name that geometry , 
entered into their study. ~~ 

So we find in Euclid’s “Elements of Geometry”’ 
(B.c., c. 300) formulae for (2+ 0) and other simple. 
algebraic relations worked out and proved by geo- , 
metric figures. Hence Euclid and his followers knew 


1 Besides Eisenlohr’s translation already mentioned, see Cantor, I, 


p. 38. Ashort sketch is given in Gow’s History of Greek Mathematics, 
Deis. 


ay 


THE GROWTH OF ALGEBRA 147 


from the figure that to “complete the square,” the 
geometric square, of 2*+ 2a, it is necessary to add 


a’, 


He also solved, geometrically, quadratic equations 
of the form ax — 27 = 6, ax +27= 46, and simultaneous 
equations of the form r+ y=a, ry= 0 

With the older Greek view of mathematics, how- 
ever, it was impossible for algebra to make much 
headway. Recognizing the linear, quadratic, and 
cubic functions of a variable, because these could be 
represented by lines, squares, and cubes, the Greeks 
of Euclid’s time refused to consider the fourth power 
of a variable because the fourth dimension was 
beyond their empirical space. 

Algebra had, however, made a beginning before 
Euclid’s time. Thymaridas of Paros, whose personal 
history is quite unknown, had already solved some 
simple equations, and had been the first to use the 
expressions g7ven or defined (wptcpévot), and unknown 
or undefined (acptorot),? and it seems not improbable 
that the quadratic equation was somewhat familiar 
before the Alexandrian school was founded? Aris- 
totle, too, had employed letters to indicate unknown 
quantities in the statement of a problem, although 
not in an equation.’ 


\ 1 Heath, T. L., Diophantos of Alexandria, Cambridge, 1885, p. 140. 
)2 Cantor, I, p. 148; Gow, p. 97, 107. 

.8 Cantor, I, p. 301 ; but see Heath’s Diophantos, p. 139. 
/ 4 Gow, p. 105. 


~ 
— 


148 THE TEACHING OF ELEMENTARY MATHEMATICS 


The most notable advance before the Christian era 
was made by Heron of Alexandria, about 100 B.c. 
Breaking away from the pure geometry of his prede- 
cessors, and not hesitating to speak of the fourth 
power of lines, he solved the quadratic equation! and 
even ran up against imaginary roots.2, This was the 
turning-point of Greek mathematics, the downfall of 
their pure geometry, the rise of a new discipline. 

But it is to Diophantus that we owe the first 
serious attempt to work out this new science. An 
Alexandrian, living in the fourth century, probably in 
the first half, he wrote a work, “ApiOuntica, almost 
entirely devoted to algebra.2 This work is the first 
one known to have been written upon algebra alone 
(or chiefly). Diophantus uses only one unknown 
quantity, o dpuOucs or o adpiotos apiOucs, symbolizing 
it by s’ or s%.4 The square he calls Svvapus, power 
(its symbol 6°), the cube «vos (x*), and he also gives 
names to the fourth, fifth, and sixth powers. He 
has symbols for equality and for subtraction, and the 
modern expression #2—527+82-—z1 he would write 


1 Cantor, I, p. 377; Gow, p. 106. 
2 Cantor, I, p. 374; Beman, W. W., vice-presidential address, Section 
«fi, American Assoc. Adv. Sci., 1897. 
Drs Heath, T. L., Diophantos of Alexandria, Cambridge, 1885; Gow, 
p. 100; Hankel and Cantor, of course, on all such names. De Morgan 
has a good article on Diophantus in Smith’s Dict. of Gk. and Rom. Biog., 
a work containing several valuable biographies of mathematicians. 
4 For discussion of the symbol, see Heath, p. 56-66. 


THE GROWTH OF ALGEBRA 149 


in the form x*as"npd%ep a a form not particularly 
more difficult than our own. The nature of his solu- 
tions will be understood from the following example, | 
modern symbols being here used: “Find two num- 
bers whose sum is 20 and the difference of whose 
squares is 80. 

Put for the numbers ++ 10, 1I0O—¥%. 

Squaring, we have 2#+ 20% + 100, «7+ 100— 20%. 

The difference, 40 + = 80. 

Dividing, cars 72 

Result, greater is 12, less is 8.”* This does not 
differ from our own present plan, although being less 
troubled by negative numbers we would probably say: 

(20 — r+ — 27 = 80. 
.. 400 — 40 4 = 80. 
ag2O — 10 7 
Pata tf. aN, 2O — 4 == 12, 

It thus appears that Diophantus understood the 
simple equation fairly well. The quadratic, however, he 
solved merely by rule. Thus he says, “842°—-747=7, 
therefore +=4,” giving but one of the two roots. 
Of the negative quantity he apparently knew nothing, 
and his work was limited, with the exception of a 
single easy cubic, to equations of the first two 
degrees. His favorite subject was indeterminate 


1 Heath, p. 72. FID. p70, 


150 THE TEACHING OF ELEMENTARY MATHEMATICS 


equations of the second degree, and on this account 
indeterminate equations in general are often desig- 
nated as Diophantine. One of the most remarkable 
facts connected with the work of Diophantus is that, 
although most other algebraists down to about 1700 
A.D., used geometric figures more or less, he nowhere 
appeals to them.t Summing up the work of the 
Greeks in this field, we may say that they could 
solve simple and quadratic equations, could represent 
geometrically the positive roots of the latter, and 
could handle indeterminate equations of the first and 
second degrees. 

Oriental algebra—It was long after the time of 
Diophantus, and in a country well removed from 
Greece, and among a race greatly differing from the 
Hellenic people, that algebra took its next noteworthy 
step forward. It is true that Aryabhatta, a Hindu 
mathematician (b. 476), made some contributions to the 
subject not long after Diophantus wrote, but he did not 
carry the subject materially farther than the Greeks,? 
and it was not until about 800 a.p. that the next real 
advance was made. 

When under the Calif Al-Mansur (the Victorious, 
c. 712-775) it was decided to build a new capital for 


1 Gow, p. 114.n.; Hankel, p. 162. 

2 Cantor, I, p. 575; Hankel, p. 172; Matthiessen, L., Grundziige der 
antiken und modernen Algebra der litteralen Gleichungen, 2. Ausg., Leip- 
zig, 1896, p. 967. 


THE GROWTH OF ALGEBRA ‘51 


the Mohammedan rulers, the site of an ancient city 
dating back to Nebuchadnezzar’s time, on the banks of 
the Tigris, was chosen. To this new city of Bagdad 
were called scholars from all over the civilized world, 
Christians from the West, Buddhists from the East, 
and such Mohammedans as might, in those early days 
of that religion, be available. With this enlightened 
educational policy, a policy opposed to in-breeding and 
to sectarianism, Bagdad soon grew to be the centre of 
the civilization of that period. Under Harun-al-Raschid 
(Aaron the Just, calif from 786 to 809) the califate 
reached the summit of its power, extending from the 
Indus to the Pillars of Hercules. His son Al-Mamun 
(786 — 833), whom Sismondi calls “the father of letters 
and the Augustus of Bagdad,’ brought Arab learning 
toits height. It was during his reign, in the first quarter 
of the ninth century, that there came from Kharezm 
(Khwarazm), a province of Central Asia, a mathemati- 
cian known from his birthplace as Al-Khowarazmi.! 
He wrote the first general work of any importance on 
algebra, that of Diophantus being largely confined to a 
single class of equations, and to the science he gave its 
present name. He designated it (Im aljabr wa’l mu- 
gabalah, that is, “the science of redintegration and 


) 


equation,’ a title which appeared in the thirteenth cen- 


tury Latin as /udus algebre almucgrabaleque, in six- 


1 Abu Ja’far Mohammed ben Musa al-Khowarazmi, Abu Ja’far Moham- 
med son of Moses from Kharezm. Cantor, I, p. 670, 


I52 THE TEACHING OF ELEMENTARY MATHEMATICS 


teenth century English as algiebar and almachabel, and 
in modern English as a/gebva.1_ So important were also 
his writings on arithmetic, that just as ‘ Euclid” is in 
England a synonym for elementary geometry, so a/go- 
ritme (from al-Khowarazmt) was for a long time a syn- 
onym for the science of numbers, a word which has 
survived in our a/gorism (algorithm). | 
Al-Khowarazmi discussed the solution of simple 
and quadratic equations in a scientific manner, dis- 
tinguishing six different classes, much as our old-style 
writers on arithmetic distinguished the various “cases” 
of percentage. His classes were, in modern notation, 
ak?=b4, arts c, bx Sc x7 bro 4 eS Or 
bx +c, showing how primitive was the science which 
could not grasp the general type ar*+ dr+c=0. 
His method of stating and solving a problem may 
be seen in the following:% “Roots and squares are 


equal to numbers; for instance, one square and ten © 


roots of the same amount to thirty-nine ;* that is to 
say, what must be the square which, when increased 
by ten of its own roots, amounts to thirty-nine? The 
solution is this: you halve the number of the roots, 
which in the present instance yields five. This you 
multiply by itself; the product is twenty-five. Add 


1 See also Heath, p. 149. 2 Cantor, I, p. 676. 

8 From The Algebra of Mohammed-ben-Musa, edited and translated 
by Frederic Rosen, London, 1831. 

£52... 47 410 4 = 80; 


eS 


THE GROWTH OF ALGEBRA 153 


this to thirty-nine; the sum is sixty-four. Now take 
the root! of this, which is eight, and subtract from 
it half the number of the root, which is five; the 
remainder is three. This is the root of the square 
for which you sought.”* The solution merely sets 
forth without explanation the rule expressed in our 
familiar formula for the solution of 27+4r+9=0, 


ne, <= — - +i ~Vp*— 4g, except that only one root 


is given. He however recognizes the existence of 
two roots where both are real and positive, as in the 
equation 27+ 21=10%.% In practice he commonly 
_ uses but one root. 

Sixteenth century algebra— Algebra made little ad- 
vance, save in the way of the solution of a few special 
cubics, from the time of Mohammed ben Musa to the 
sixteenth century, seven hundred years. Its course 
had run from Egypt to Greece, and from Greece (and 
Grecian Alexandria) to Persia. It now transfers itself 
from Persia to Italy and works slowly northward. 

In a famous work frinted in Niirnberg in 1545, 
the “Ars magna,’’* Cardan gives a complete solution of 
a cubic equation; that is, he solves an equation of the 


1 7.e., the square root. 

2The successive steps are as follows: } of lO=5; 5-5 = 25; 25 
+ 39=64; V64= 8; 8—5 =3. 

8 Rosen, p. II. 

*Hieronymi Cardani, prastantissimi mathematici, philosophi, ac 
medici, Artis Magne, sive de regvlis algebraicis, Lib. unus. 


154. THE TEACHING OF ELEMENTARY MATHEMATICS 


form #°+pr= 49, to which all other cubics can be 
reduced. He mentions, however, his indebtedness to 
earlier writers, though not as generously as seems to 
have been their due.! | 

This is not the place to consider the relative claims 
of Cardan, Tartaglia (Tartalea) Ferro (Ferreus), 
and Fiori (Florido). Cardan seems to have obtained 
Tartaglia’s solution of the cubic under pledge of 
secrecy and then to have published it. But however 
this was, by the middle of the sixteenth century the 
cubic equation was solved, and Ludovico Ferrari at 
about the same time solved the quartic. 

Algebra had now reached such a point that mathema- 
ticlans were able to solve, in one way or another, gene- 
ral equations of the first four degrees. Thereafter the 
chief improvements were (1) in symbolism, (2) in under- 
standing the number system of algebra, (3) in finding 
approximate roots of higher numerical equations, (4) in 
simplifying the methods of attacking equations, and (5) 
in the study of algebraic forms. For the purposes of 
elementary algebra we need at this time to speak only 
of the first three. 


1 Scipio Ferreus Bononiensis iam annis ab hinc triginta ferme capit~ 
ulum hoc inuenit, tradidit uero Anthonio Marize Florido Veneto, qui 
cii in certamen citi Nicolao Tartalea Brixellense aliquando uenisset, 
occasionem dedit, ut Nicolaus inuenerit, & ipse, qui cum nobis rogan- 
tibus tradidisset, suppressa ademonstratione, freti hoc auxilio, demonstra- 
tionem quesiuimus, eamque in modos, quod difficillimum fuit, redactam 
sic subiecimus. Fol. 29, v. 


THE GROWTH OF ALGEBRA 155 


Growth of symbolism — Algebra, as is readily seen, is 
very dependent upon its symbolism. Its, history has 
been divided into three periods, of rhetorical, of synco- 
pated, and of symbolic algebra. The rhetorical algebra 
is that in which the equation is written out in words, as 
in the example given on p. 152 from Al-Khowarazmi; 
the syncopated, that in which the words are abbre- 
viated, as in most of the example given on p. 149 
from Diophantus; the symbolic, that in which an 
arbitrary shorthand is used, as in our common algebra 
of to-day. 

The growth of symbolism has been slow. From the 
radical sign of Chuquet (1484), R*. 10, through various 
other forms, as +/33 10, to our common symbol, Vv 10 
and to the more refined 10°, which is only slowly becom- 
ing appreciated in elementary schools, is a tedious and a 
wandering path. So from Cardan’s 


cubus p 6. rebus zequalis 20, for 2°+6 += 20, 
through Vieta’s 
IC —8Q+ 16N equ. 40, for #7 — 827+ 164=40, 
and Descartes’s 
x? » axr— bb, for ?7=ar— 6, 


and Hudde’s 
2° 0 gx.7, for A=gr-+7,! 


1Beman and Smith’s translation of Fink’s History of Mathematics, 
p- 108, 


156 THE TEACHING OF ELEMENTARY MATHEMATICS 


has likewise been a long and tiresome journey. Such > 
simple symbols as the x for multiplication,! and the still 
simpler dot used by Descartes, the = for equality,” 


the *-for He these all had a long struggle for recog- 
os 


nition. Even now the symbol + has only a limited 
acceptance in the mathematical world, and there are 
three widely used forms for the decimal point.* Thus 
symbolism has been a subject of slow growth, and we 
are still in the period of unrest. 

We may, however, assign to the Frenchman Vieta® 
the honor of being the founder of symbolic algebra in 
large measure as we recognize it to-day. His first book 
on algebra, “In artem analyticam isagoge,” appeared in 
1591. Laisant thus summarizes his contribution: “He 
it is who should be looked upon as the founder of alge- 
bra as we conceive it to-day. The powerful impulse 
which he gave consisted in this, that while unknown 
quantities had already been represented by letters to 
facilitate writing, it was he who applied the same method | 
to known quantities as well. From that day, when the 
search for values gave way to the search for the opera- 
tions to be performed, the idea of the mathematical 


1 First used by Oughtred in 1631. 

2 Recorde, 1556. 8 Wallis. 

4214.is usually written 2.5 in America, 2-5 in England, 2,5 on the Con- 
tinent. 

5 Francois Viéte, 1540-1603. 

6 Cantor, II, p. 577; for a general summary of his work, see p. 595. 


THE GROWTH OF ALGEBRA 157 


function enters into the science, and this is the source of 
its subsequent progress.” ! 

Number systems— The difficulty of understanding 
the number systems of algebra has been, perhaps, the 
greatest obstacle to its progress. The primitive, 
natural number is the positive integer. So long as 
the world met only problems which may be repre- 
sented by the modern form ar+6=c, where c>d 
and c— 0 is a multiple of a, as in 37+2= TIT, these 
numbers sufficed. But when problems appeared which 
involve the form of equation ar= 6 where @ is not 
memulipic of @, as-in 3.4%— 1, or 2, or 5, then: other. 
kinds of number are necessary, the unit fraction, the 
general proper fraction, and the improper fraction or 
mixed number. We have seen (Chap. III) how the 
world had to struggle for many centuries before it came 
to understand numbers of this kind. It was only by 
an appeal to graphic methods (the representation of 
numbers by lines) that the fraction came to be under- 
stood. When, further, problems requiring the solution 
of an equation like +” =a, a not being an 2” power, as 
in #7 = 2, still a new kind of number was necessary, the 
real and irrational number, a form which the Greeks 
interpreted geometrically for square and cube roots. 

The next step led to equations like r+a= 4, with 
a>o,asin*x+5 = 2,a form which for many centuries 
baffled mathematicians because they could not bring 


1a Mathématique, p. 55. 


158 THE TEACHING OF ELEMENTARY MATHEMATICS 


themselves to take the step into the domain of nega- 
tive numbers. It was not until the genius of Des- 
cartes (1637) more completely grasped the idea of 
the one-to-one correspondence between algebra and 
geometry, that the negative number was taken out 
of the domain of xumere ficte! and made entirely 
real. One more step was, however, necessary for 
the solution of equations of the form 2#”+a=o. 
What to do with an equation like 27+ 4=0 was still 
an unanswered question. To say that += V— 4, or 
2V—1, or +2V—1, avails nothing unless we know 
the meaning of the symbol “V—1.” It was not 
until the close of the eighteenth century that any 
considerable progress was made in the interpretation 
of the symbol a+4V—1. In 1797 Caspar Wessel, 
a Norwegian, suggested the modern interpretation, and 
published a memoir upon complex ‘numbers in the 
proceedings of the Royal Academy of Sciences and 
Letters of Denmark for 1797.2. Not, however, until 
Gauss published his great memoir on the subject 
(1832) was the theory of the graphic representation of 


1 Cardan, Ars magna, 1545, Fol. 3, v. 

2 This has recently been republished in French translation, under the 
title Essai sur la réprésentation analytique de la direction, Copenhague, 
1897, with a historical preface by H. Valentiner. For a valuable summary 
of the history, see the vice-presidential address of Professor Beman, Section 
A of the American Assoc. Adv. Sci., 1897. A brief summary is also 
given in the author’s History of Modern Mathematics, in Merriman and 
Woodward’s Higher Mathematics, New York, 1896. 


THE GROWTH OF ALGEBRA 159 


the complex number generally known to the mathemat- 
ical world. Elementary text-book writers still seem 
indisposed to give the subject place, although its 
presentation is as simple as that of negative numbers.! 

For the purposes of elementary teaching only a 
single other historical question demands consideration, 
the approximate solution of numerical equations, and 
even this is rather one of arithmetic than of algebra. 
Algebra has proved that there is no way of solving 
the general equation of degree higher than four; that 
is, that by the common operations of algebra we can 
solve the equation | 

ax*+ bx? + cr2*+dxr+e=0, 
but that we cannot solve the equation 
ax®+bxt+eri+dP+exr+f=0- 
We can, however, approximate the real roots of any 
numerical algebraic equation, and this suffices for 
practical work. That is, we can find that one root of 
the equation : 
a°4 1274+ 59434 15042+ 210%— 207 =0 

is 0.638605803 +, 
but we have no formula for solving such equations by 
algebraic operations as we have for solving 


a? + pxrt+gq=0. 


1 For an elementary treatment, see Beman and Smith’s Algebra, Boston, 
1900. 

2 For historical résumé, see the author’s History of Modern Mathematics 
already cited, p, 519. 


160 THE TEACHING OF ELEMENTARY MATHEMATICS 


The simple method now generally used for this ap- 
proximation is due to an Englishman, W. G. Horner, 
who published it in 1819, and it now appears in ele- 
mentary works in English as ‘ Horner’s method.” 
Foreign writers have, however, been singularly slow 


in recognizing its value. 


CHAPTER VII 
ALGEBRA, — WHAT AND WHy TAUGHT 


Algebra defined—-In Chapter VI the growth of 
‘algebra was considered in a general way, assuming 
that its nature was fairly well known. Nor is it 
without good reason that this order was taken, for 
the definition of the subject is best understood when 
considered historically. But before proceeding to dis- 
cuss the teaching of the subject it is necessary to 
examine more carefully into its nature. 

It is manifestly impossible to draw a definite line be- 
tween the various related sciences, as between botany 
and zodlogy, between physics and astronomy, between 
algebra and arithmetic, and so on. The child who 
Meets, the expression 2 X.(?)= 8, in the first (grade, 
has touched the elements of algebra. The student of 
algebra who is called upon to simplify 

(2+ V3)/(2- V3) 
is facing merely a problem of arithmetic. In fact, 
a considerable number of topics which are _ prop- 
erly parts of algebra, as the treatment of propor- 
tion, found lodgment in arithmetic before its sister 
science became generally known; while much of 


arithmetic, like the theory of irrational (including 
M 161 


162 THE TEACHING OF ELEMENTARY MATHEMATICS 


complex) numbers, has found place in algebra simply 
because it was not much needed in practical arith- 
metic.1 

Recognizing this laxness of distinction between the 
two sciences, Comte? proposed to define algebra ‘“‘as 


having for its object the resolution of equations; 


taking this expression in its full logical meaning, 
which signifies the transformation of implicit func- 
tions into equivalent explicit ones.2 In the same way 
arithmetic may be defined as destined to the deter- 
mination of the values of functions. Henceforth, 
therefore, we will briefly say that Algebra is the 
Calculus of Functions, and Arithmetic the Calculus of 
Values.” 4 

Of course this must not be taken as a definition 
universally accepted. As a prominent writer upon 
“methodology” says: “It is very difficult to give a 


1 Teachers who care to examine one of the best elementary works upon 
arithmetic in the strict sense of the term, should read Tannery, Jules, 
Legons d’Arithmétique théorique et pratique, Paris, 1894. 

2The Philosophy of Mathematics, translated from the Cours de Philo- 
sophie positive, by W. M. Gillespie, New York, 1851, p. 55. 

8 Je, in x2 + px + g =O we have an implicit function of 4 equated to 
zero ; this equation may be so transformed as to give the explicit function 


4 
x=— 5+} P2-—44; 


and this transformation belongs to the domain of algebra. 

4 Laisant begins his chapter L’Algébre (La Mathématique, p. 46) by 
reference to this definition, and makes it the foundation of his discussion 
of the science. 


ALGEBRA, — WHAT AND WHY TAUGHT 163. 


good definition of algebra. We say that it is merely 
a generalized or universal arithmetic, or rather that 
‘it is the science of calculating magnitudes con- 
sidered generally’ (D’Alembert). But as Poinsot has 
well observed, this is to consider ‘it under a point of 
view altogether too limited, for algebra has two 
distinct parts. The first part may be called universal 
arithmetic. ... The other part rests on the theory 
of combinations and arrangement. ... We may 
give the following definition. ... Algebra has for its 
object the generalizing of the solutions of problems 
relating to the computation of magnitudes, and of 
studying the composition and transformations of for- 
mulae to which this generalization leads.”! The best 
of recent English and French elementary algebras 
make no attempt at defining the subject.? 

The function — Taking Comte’s definition as a point 
of departure, it is evident that one of the first steps 
in the scientific teaching of algebra is the fixing of 
the idea of function. How necessary this is, apart 
from all question of definition, is realized by all 
advanced teachers. “I found,’ says Professor Chrys- 
tal, “when I first tried to teach university students 
coordinate geometry, that I had to go back and 


1 Dauge, Félix, Cours de Méthodologie Mathématique, 2. éd., Gand et 
Paris, 1896, p. 103. 

2 Chrystal, G., 2 vols. 2 ed., Edinburgh, 1889. Bourlet, C., Legons 
d’Algébre élémentaire, Paris, 1896. 


164 THE TEACHING OF ELEMENTARY MATHEMATICS 


teach them algebra over again. The fundamental 
idea of an integral function of a certain degree, 
having a certain form and so many coefficients, was 
to them as much an unknown quantity as the pro- 
Verpial cons} 

Happily this is not only pedagogically one of the 
first steps, but practically it is a very easy one 
because of the abundance of familiar illustrations. 
“Two general circumstances strike the mind; one, 
that all that we see is subjected to continual trans- 
formation, and the other that these changes are 
mutually interdependent.” Among the best elemen- 
tary illustrations are those involving time; a stone 
falls, and the distance varies as the time, and vwéce 
versa, we call the distance a function of the time, 
and the time a function of the distance. We take a 
railway journey; the distance again varies as the 
time, and again time and distance are functions of 
each other. Similarly, the interest on a note is a 
function of the time, and also of the rate and the 
principal. 

This notion of function is not necessarily foreign 
to the common way of presenting algebra, except 
that here the idea is emphasized and the name is 
made prominent. Teachers always give to beginners 
problems of this nature: Evaluate 27+ 22x+1 for 
4 = 2, 3, etc., which is nothing else than finding the 


1 Presidential address, 1885. 2 Laisant, p. 46. 


ALGEBRA,— WHAT AND WHY TAUGHT 165 


value of a function for various values of the variable. 
Similarly, to find the value of a2 +3a70+3a?+ 6 
for @=1, 0=2, is merely to evaluate a certain func- 
tion of @ and 4%, or, as the mathematician would say, 
J (a, 6), for special values of the variables. It is 
thus seen that the emphasizing of the nature of the 
function and the introduction of the name and the 
symbol are not at all difficult for beginners, and they 
constitute a natural point of departure. The introduc- 
tion to algebra should therefore include the giving of 
values to the quantities which enter into a function, 
and thus the evaluation of the function itself. 

Having now defined algebra as the study of certain 
functions,! which includes as a large portion the solution 
of equations, the question arises as to its value in the 
curriculum. 

Why studied — Why should one study this theory of 
certain simple functions, or seek to solve the quadratic 
equation, or concern himself with the highest common 
factor of two functions? It is the same question which 
meets all branches of learning, —cwz dono? Why should 
we study theology, biology, geology — God, life, earth ? 
What doth it profit to know music, to appreciate Pheid- 
jas, to stand before the fagade at Rheims, or to wonder 


1 Certain functions, for functions are classified into algebraic and trans- 
cendental, and with the latter elementary algebra concerns itself but little. 
£.g., algebra solves the algebraic equation «4 = 4, but with the transcen- 
dental equation a* = 4 it does not directly concern itself. 


166 THE TEACHING OF ELEMENTARY MATHEMATICS 


at the magic of Titian’s coloring? As Malesherbes 
remarked on Bachet’s commentary on Diophantus, “It 
won’t lessen the price of bread;”! or as D’Alembert 
retorts from the mathematical side, @ propos of the Iphi- 
eénie of Racine, “ What does this prove?” 

Professor Hudson has made answer: “To pursue an 
intellectual study because it ‘pays’ indicates a sordid 
spirit, of the same nature as that of Simon, who wanted 
to purchase with money the power of an apostle. The 
real reason for learning, as it is for teaching algebra, is, 
that it is a part of Truth, the knowledge of which is its 
own reward. 

“Such an answer is rarely satisfactory to the ques- 
tioner. He or she considers it too vague and too wide, 
as it may be used to justify the teaching and the learn- 
ing of any and every branch of truth; and so, indeed, it 
does. A true education should seek to give a knowledge 
of every branch of truth, slight perhaps, but sound as 
far as it goes, and sufficient to enable the possessor to 
sympathize in some degree with those whose privilege it 
is to acquire, for themselves at least, and it may be for 
the world at large, a fuller and deeper knowledge. A 
person who is wholly ignorant of any great subject of 
knowledge is like one who is born without a limb, and 
is thereby cut off from many of the pleasures and inter- 
ests of life. 

1«Te commentaire de Bachet sur Diophante ne fera pas diminuer le 
prix du pain.” 


ALGEBRA, — WHAT AND WHY TAUGHT 167 


“T maintain, therefore, that algebra is not to be taught 
on account of its utility, not to be learnt on account of 
any benefit which may be supposed to be got from it;. 
but because it is a part of mathematical truth, and no 
one ought to be wholly alien from that important depart- 
ment of human knowledge.”’! 

The sentiments expressed by Professor Hudson will 
meet the approval of all true teachers. Algebra is 
taught but slightly for its utilities to the average citizen. 
-Useful it is, and that to a great degree, in all subsequent 
mathematical work; but for the merchant, the lawyer, 
the mechanic, it is of slight practical value. 

Training in logic— But Professor Hudson states, in 
the above extract, only a part of the reason for teaching 
the subject — that we need to know of it as a branch of 
human knowledge. This might permit, and sometimes 
seems to give rise to, very poor teaching. We need it 
also as an exercise in logic, and this gives character to 
the teacher’s work, raising it from the tedious, barren, 
mechanical humdrum of rule-imparting to the plane of 
true education. Professor Hudson expresses this idea 
later in his paper when he says, “Rules are always 
faischievous so long as they are necessary: it is only 
when they are superfluous that they are useful.” 

Thus to be able to extract the fourth root of 444+423 
+622+4%7+1 is a matter of very little moment. The 


1Hudson, W. H. H., On the Teaching of Elementary Algebra, paper 
before the Educational Society (London), Nov. 29, 1886. 


168 THE TEACHING OF ELEMENTARY MATHEMATICS 


pupil cannot use the result, nor will he be liable to use 
the process in his subsequent work in algebra. But 
that he should have power to grasp the logic involved 
in extracting this root is very important, for it is this 
very mental power, with its attendant habit of concen- 
tration, with its antagonism to wool-gathering, that we 
should seek to foster. To have a rule for finding the 
highest common factor of three functions is likewise a 
matter of little importance, since the rule will soon fade 
from the memory, and in case of necessity a text-book 
can easily be found to supply it; but to follow the logic 
of the process, to keep the mind intent upon the opera- 
tion while performing it, herein lies much of the value 
of the subject,—here is to be sought its chief vazsoz 
a etre. 

Hence the teacher who fails to emphasize the idea 
of algebraic function fails to reach the pith of the 
science. The one who seeks merely the answers to 
a set of unreal problems, usually so manufactured 
as to give. rational results alone, instead of seeking to 
give that power which is the chief reason for alge- 
bra’s being, will fail of success. It is of little value 
in itself that the necessary and sufficient condition 
for 2—a2=—=0 is that += 0; t=194 = — 2 putes 
“of great value to see why this is such condition. 

Practical value— Although for most people algebra 
is valuable only for the culture which it brings, at the 
same time it has never failed to appeal to the common 


ALGEBRA, — WHAT AND WHY TAUGHT 169 


sense of practical men as valuable for other reasons. 
All subsequent mathematics, the theory of astronomy, 
of physics, and of mechanics, the fashioning of guns, 
the computations of ship building, of bridge building, 
and of engineering in general, these rest upon the opera- 
tions of elementary algebra. Napoleon, who was nota 
man to overrate the impractical, thus gave a statesman’s 
estimate of the science of which algebra is a corner- 
stone: “The advancement, the perfecting of mathe- 
matics, are bound up with the prosperity of the State.” ? 

Ethical value — There are those who make great claims 
for algebra, as for other mathematical disciplines, as 
a means of cultivating the love for truth, thus giving 
to the subject a high ethical value. Far be it from 
teachers of the science to gainsay all this, or to antago- 
nize those who follow Herbart in bending all education 
to bear upon the moral building-up of the child. But 
we do well not to be extreme in our claims for mathe- 
matics. Cauchy, one of the greatest of the French 
mathematicians of the nineteenth century, has left us 
some advice along this line: ‘‘ There are other truths 
than the truths of algebra, other realities than those of 
sensible objects. Let us cultivate with zeal the mathe- 
matical sciences, without seeking to extend them beyond 
their own limits; and let us not imagine that we can 
attack history by formulae, or employ the theorems of 


1 L’avancement, le perfectionnement des mathématiques sont liés a la 
prospérité de |’ Etat. 


170 THE TEACHING OF ELEMENTARY MATHEMATICS 


algebra and the integral calculus in the study of ethics.” 
For illustration, one has but to read Herbart’s Psychology 
to see how absurd the extremes to which even a great 
thinker can carry the applications of mathematics. 

Of course algebra has its ethical value, as has every 
subject whose aim is the search for truth. But the 
direct application of the study to the life we live is very 
slight. When we find ourselves making great claims 
of this kind for algebra, it is well to recall the words 
of Mme. de Staél, paying her respects to those who, in 
her day, were especially clamorous to mathematicize all 
life: ‘““Nothing is less applicable to life than mathe- 
matical reasoning. A proposition in mathematics is 
decidedly false or true; everywhere else the true is 
mixed in with the false.” 

When studied — Having framed a tentative defini- 
tion of algebra, and having considered the reason for 
studying the science, we are led to the question as 
to the place of algebra in the curriculum. 

At the present time, in America, it is generally 
taken up in the ninth school year, after arithmetic 
and before demonstrative geometry. Since most 
teachers are tied to a particular local school system, 
as to matters of curriculum, the question is not to 
them a very practical one. But as a problem of 
education it has such interest as to deserve attention. 

Quoting again from Professor Hudson: “The be- 
ginnings of all the great divisions of knowledge 


ALGEBRA,— WHAT AND WHY TAUGHT 17! 


should find their place in a perfect curriculum of 
education ; at first something of everything, in order 
later to learn everything of something. But it is 
needless to say all subjects cannot be taught at once, 
all cannot be learnt at once; there is an order to be 
observed, a certain sequence is necessary, and it may 
well be that one sequence is more beneficial than an- 
other. My opinion is that, of this ladder of learning, 
Algebra should form one of the lowest rungs; and I 
find that in the Mznxetcenth Century for October, 1886, 
the Bishop of Carlisle, Dr. Harvey Goodwin, quotes 
Comte, the Positivist Philosopher, with approval, to 
the same effect. 

‘The reason is this: Algebra is a certain science, 
it proceeds from unimpeachable axioms, and its con- 
clusions are logically developed from them; it has its 
own special difficulties, but they are not those of 
weighing in the balance conflicting probable evidence 
which requires the stronger powers of a maturer 
mind. It is possible for the student to plant each 
step firmly before proceeding to the next, nothing is 
left hazy or in doubt; thus it strengthens the mind 
and enables it better to master studies of a different 
nature that are presented to it later. Mathematics 
give power, vigor, strength, to the mind; this is 
commonly given as the reason for studying them. I 
give it as the reason for studying Algebra early, that 
‘is to say, for beginning to study it early; it is not 


172 THE TEACHING OF ELEMENTARY MATHEMATICS 


necessary, it is not even possible, to finish the study 
of Algebra before commencing another. On the 
other hand, it is not necessary to be always teaching 
Algebra; what we have to do, as elementary teachers, 
is to guide our pupils to learn enough to leave the 
door open for further progress; we take them over 
the threshold, but not into the innermost sanctuary. 

“The age at which the study of Algebra should 
begin differs in each individual case. ... It must be 
rare that a child younger than nine years of age is 
fit to begin; and although the subject, like most 
others, may be taken up at any age, there is no 
superior limit; my own opinion is, that it would be 
seldom advisable to defer the commencement to later 
than twelve years.” ~- 

This opinion has been quoted not for indorsement, 
but rather as that of a teacher and a mathematician 
of such prominence as to command respect. The 
idea is quite at variance with the American custom 
of beginning at about the age of fourteen or fifteen, 
or even later, and it raises a serious question as to 
the wisdom of our course. Indeed, not only is the 
question of age involved, but also that of general 
sequence. Are we wise in teaching arithmetic for 
eight years, dropping it and taking up algebra, drop- 
ping that and taking up geometry, with possibly a 
brief review of all three later, at the close of the. 
high school course? 


ALGEBRA,— WHAT AND WHY TAUGHT 173 


Fully recognizing the folly of a dogmatic state- 
ment of what is the best course, and hence desiring 
to avoid any such statement, the author does not 
hesitate to express his personal conviction that the 
present plan is not a wisely considered one. He 
feels that with elementary arithmetic should go, as 
already set forth in Chapter V, the simple equation,} 
and also metrical geometry with the models in hand; 
that algebra and arithmetic should run side by side 
during the eighth and ninth years, and that demon- 
strative geometry should run side by side with the 
latter part of algebra. One of the best of recent 
series of text-books, Holzmiiller’s,? follows this general 
plan, and the arrangement has abundant justification 
in most of the Continental programmes. It is so scien- 
tifically sound that it must. soon find larger acceptance 
in English and American schools. 

Arrangement of text-books— As related to the sub- 
ject just discussed, a word is in place concerning the 
arrangement of our text-books. It is probable that 
we shall long continue our present general plan of 
having a book on arithmetic, another on algebra, and 
still another on geometry, thus creating a mechanical 
barrier between these sciences. We shall also, doubt- 


1 There is a good article upon this by Oberlehrer Dr. M. Schuster, Die 
Gleichung in der Schule, in Hoffmann’s Zeitschrift, XXIX. Jahrg. (1898), 
p. 81. 

2 Leipzig, B. G. Teubner. 


174 THE TEACHING OF ELEMENTARY MATHEMATICS 


less, combine in each book the theory and the exer- 
cises for practice, because this is the English and 
American custom, giving in our algebras a few pages 
of theory followed by a large number of exercises. 
The Continental plan, however, inclines decidedly 
toward the separation of the book of exercises from 
the book on the theory, thus allowing frequent 
changes of the former. It is doubtful, however, if 
the plan will find any favor in America, its advan- 
tages being outweighed by certain undesirable fea- 
tures.1 There is, perhaps, more chance for the adoption 
of the plan of incorporating the necessary arithmetic, 
algebra, and geometry for two or three grades into 
a single book, a plan followed ‘by Holzmiiller with 
much success. 


1 An interesting set of statistics with respect to German text-books is 
given by J. W. A. Young in Hoffmann’s Zeitschrift, XXIX. Jahrg. (1898), 
p- 410, under the title, Zur mathematischen Lehrbiicherfrage. 


CHAPTER VIII 
TYPICAL PARTS OF ALGEBRA 


Outline — While it is not worth while in a work of 
this kind to enter into commonplace explanations of 
matters which every text-book makes more or less 
lucid, it may be of value to call attention to certain 
topics that are somewhat neglected by the ordinary 
run of classroom manuals. The teacher is depend- 
ent upon his text-book for most of his exercises, 
since the dictation of any considerable number is a 
waste of time. He is likewise dependent upon the 
book for much of the theory, since economy of time 
and of students’ effort requires him to follow the 
text unless there is some unusual reason for depart- 
ing from it. But he is not dependent upon the book 
for the sequence of topics, nor for all of the theory, 
nor for all of his problems; neither is he precluded 
from creating all the interest possible, and introduc- 
ing a flood of light, through his superior knowledge of 
the subject. For this reason this chapter is written, 
that it may add to the teacher’s interest by throwing 
some light upon a few typical portions, and may 
suggest thereby some improved methods of treating 
the entire subject. 

175 


176 THE TEACHING OF ELEMENTARY MATHEMATICS 


Definitions — The policy of learning any consider- 
able number of definitions at the beginning of a new 
subject of study has already been discussed in Chap- 
ter II. The idea is always of vastly more impor- 
tance than the memorized statement. At the same 
time there is much danger from the inexact defini- 
tions to be found in many text-books, a danger all 
the greater because of the pretensions of the science 
to be exact, and because there will always be found 
teachers who believe it their duty to burn the defini- 
tions indelibly into the mind. 

Whether the definitions are learned verbatim or 
not, the teacher at least will need to know whether 
they are correct. For this purpose he will find little 
assistance from other elementary school-books. He 
will need to resort to such works as Chrystal,! as 
Oliver, Wait, and Jones,? or as Fisher and Schwatt? 
in English, as Bourlet* in French, as the convenient 
little handbooks of the Sammlung Géschen® or the 
new Sammlung Schubert® in German, and Pincherle’s 
little Italian handbooks.’ 

1 Algebra, 2 vols., 2 ed., Edinburgh, 1889. 

2 A Treatise on Algebra, Ithaca, N. Y., 1887. 

8 Text-book of Algebra, part i, Philadelphia, 1898. 

* Lecons d’Algébre élémentaire, Paris, 1896. 

5 As Schubert’s Arithmetik und Algebra, and Sporer’s Niedere Analysis. 

® As Schubert’s Elementare Arithmetik und Algebra, and Pund’s Alge- 
bra, Determinanten und elementare Zahlentheorie, both published in 1899. 


7 Algebra elementare, and Algebra complementare. A good bibliog- 
raphy of this subject, for teachers, is given by T. J. McCormack in his 


TYPICAL PARTS OF ALGEBRA 177, 


A few illustrations of the general weakness of the 
common run of definitions may be of service in the 
way of leading teachers to a more critical examina- 
tion of such statements. 

The usual definition of degree of a monomial is so 
loosely stated that the beginner thinks and continues 
to think of 3 a%r3 as of the fifth degree, which it is 
in @ and x, but for the purposes of algebra, es- 
pecially in dealing with equations, it is quite as often 
considered as of the third degree in x, a distinction 
usually ignored until the student, after much stum- 
bling, comes upon it. 

A square root is usually defined as one of the two 
equal factors of an expression, although the student 
is taught, almost at the same time, that the expres- 
sion of which he is extracting the square root has 
no two equal factors. &.g., he speaks of the square 
root of #?+ 1, and yet says that 2*+ 1 is prime. 

Even so simple a concept as that of equation is 
usually defined in a fashion entirely inexpressive of 
the present algebraic meaning. Some books follow 
an anciert practice of avoiding the difficulty by 
introducing the expression ‘equation of condition,” 
and never referring to it again! In the algebra of 
to-day an equation is an equality which exists only 
for particular values of certain letters called the 
notes to the new edition of De Morgan’s work, On the Study of Mathe- 


matics, Chicago, 1898, p. 187. 
N 


178 THE TEACHING OF ELEMENTARY MATHEMATICS 


unknown quantities. As the term is used by alge- 
braists of the present time, 2+3=5 is not an equa- 
tion strictly speaking, although it expresses equality ; 
neither is a2+6=6+4a%, although it is an identity. 
An equation, as the word is now used, always con- 
tains an unknown quantity.! ! 

The term “axiom” is subject to similar abuse. No 
mathematician now defines it as “a self-evident truth,” 
and no psychology would now sanction such an unsci- 
entific statement. Algebraists, those who make the 
science to-day, agree that an axiom is merely a general 
statement so commonly accepted as to be taken for 
granted, and a statement which needs to be considered 
with care in the light of the modern advancement of the 
science. For example, no student who thinks would 
say that it is “self-evident” that “like roots of equals 
are equal.” If 4=4, it is not “self-evident” that a 
square root of 4 equals a square root of 4, for +2 does 
not equal —2. 

_Again, of what value is it to a pupil to learn the ordi- 
nary definition of addition? ‘Text-books commonly say, 
in substance, that the process of uniting two or more 
expressions in a single expression is called addition ; 
but what is meant by this “uniting”? Either the defi- 
nition would better be omitted, or it would better have, 
some approach to scientific accuracy; the choice of. 


1 De Morgan’s use of the word is not that of modern writers. See The} 


Study of Mathematics, 2 ed., Chicago, 1898, p. 57, 91. ' 
| 


TYPICAL PARTS OF ALGEBRA 179 


these alternatives may depend upon the class, or pos- 
sibly upon the teacher. 

The simple concept of factor, so vital to the pupil’s 
progress in algebra, usually suffers with the rest. Isa 
factor, as we so often read, one of several numbers or 
expressions which multiplied together make a given 
expression? In other words, is it an expression which 
will divide another? If so, are Vr+1 and Vx—r fac- 
tors of x—1? Possibly it will be said that we are limited 
to rational terms in x If so, when we ask a pupil to 
factor #®—1, shall we expect him to say that 2-1= 
(x—1)(x+h+4v —3)(4+4$—-4V—3)? This does not 
involve any irrational term in x. But possibly we are 
expected to exclude irrational and imaginary numbers 
altogether. What, then, shall we say about factoring 
w2—i? Are the factors x+4 and x—4, or are fractions 
also excluded? Is 2#—a factorable, we not knowing in 
advance but that @=4 or 9 or some other square? 
These are not trivial “catch”’ questions. Upon the 
answers depends the entire notion of factoring, the 
basis upon which we are to build the greatest part of 
algebra — the theory of equations. 

Of less importance, but still of value, is the definition 
of highest common factor. . What is the highest common 
factor of 2(a?—0*) and 4(4?—a*)? Is it 2(a—d), or 
2(d—a), or simply +(a—6)? And similarly, what is 
the lowest common multiple of a—d and d—a? These 
questions should not be puzzling; the information is 


180 THE TEACHING OF ELEMENTARY MATHEMATICS 


often needed in the simple reduction of ordinary frac- 
tions; and yet our common definitions do not throw 
much light upon them. 

The unnecessary and ill-defined term “surd”’ still clings 
to our algebras. Is it a synonym for irrational number? 
If so, what is an irrational number? Is it a number not 
rational, say V2, Va;-V—1? Is it w= 3.14150-:+,-01 
the circulate 0.666::-? Is it a single expressed root like 
/2,-0r is 24+V2-a surd?-or V2+¥3? ‘or V2+v3? 
If it is merely an irrational number, is log 2 a surd? 
These are all common expressions, arithmetical rather 
than algebraic, it is true, but conventionally holding a 
place in algebra. 

In this connection the wonder may be expressed 
as to how long we shall continue to use the terms 
“pure” and “affected” (in England adfected) quadrat- 
ics, instead of the more scientific adjectives “incom- 
plete” and “complete.” 

The inquiry might be extended much farther, but 


enough has been suggested to show the necessity | 


for care in the common definitions of algebra.! 

The awakening of interest in the subject, the vital 
point in all teaching, is best accomplished through the 
early introduction of the equation. As soon as the 


1 For those who have not access to the works mentioned on p. 176, it 
may be of service to refer to Beman and Smith’s Algebra, Boston, 1900, 
in which the authors have endeavored to state the necessary definitions 
with some approach to scientific accuracy. 


ee ee 


TYPICAL PARTS OF ALGEBRA 181 


pupil can evaluate a few functions, thus becoming 
familiar with the alphabet of algebra, the equation 
should be introduced with this object prominently in 
the teacher’s mind. 

The mere solution of the simple equation which 
the pupil first meets presents no difficulty. The 
teacher will do well to avoid such mechanical phrases 
as “clear of fractions’? and “transpose” until the 
reasoning is mastered; indeed, it may be questioned 
whether these phrases are ever of any value. Rather 
should the processes stand out strongly, thus: — 


Given ~+3= 7, to find she value of x. 
2 
Subtracting 3 from each member, = 4, 


Multiplying each member by 2, x= 8. 
To prove this (check the result), put 8 for +; 


then =+3=4+3=7, 


But the greatest difficulty which pupils have at 
this time comes from the statement of the conditions 
in algebraic language. Fortunately there is no gen- 
eral method of stating all equations, so that the pupil 
is forced out of the field of traditional rules into that 
of thought. The following outline, however, is usually 
of value in arranging the statement :— 

1. What shall x represent? In general, may be 
‘taken to represent the number in question. Z.g., in 


182 THE TEACHING OF ELEMENTARY MATHEMATICS 


the problem, “The difference of two numbers is 40 
and the sum is 450, what is the smaller number?” 
Here x (or some other such symbol) may best be 
taken to represent ‘the smaller number.” 

2. For what number described in the problem may 
two expressions be found? ‘Thus in the above prob- 
lem, the larger number is evidently 50 — x, and hence 
two expressions may be found for the difference, viz., 
40; and 50-4 — #. 

3. How do you state the equality of these expres- 
stons in algebraic language ? 

50 —+—x= 40]! 

With these directions, thus outlining a logical se- 
quence for the pupil, the statements usually offer 
little difficulty. 

Signs of aggregation often trouble a pupil more 
than the value of the subject warrants. The fact 
is, In mathematics we never find any such compli- 
cated concatenations as often meet the student almost 
on the threshold of algebra. Nevertheless the sub- 
ject consumes so little time and is of so little diffi- 
culty as hardly to justify any serious protest. Two 
points may, however, be mentioned as typical. 

First, it is a waste of time, and often a serious 
waste, to require classes to read aloud expressions like 


a+(b—2)2— {b—[a+b(b—a+c22)2— (a—2)3] —c}. 


1 Beman and Smith, Algebra. 


TYPICAL PARTS OF ALGEBRA 183 


There is no value in such an exercise in oral reading. 
Mathematicians, if by strange chance they should 
meet such an array of symbols, would never think 
of reading it aloud. Such a notion, frittering away 
time and energy and interest, is allied to that which 
labors to have —a called “negative @” instead of 
“minus @,’ which frets about ‘‘@ divided by 0” being 
called ‘“‘a@ over 6” (a mathematical expression well 
recognized by the best writers and teachers in several 
languages), and which objects to calling a” “a to 
the minus zth power” Cforgetful that mznus and 
power have long since broadened their primitive 
meaning)-— petty nothings born of the narrow views 
of some schoolmaster. 

The second point refers to a rule which still finds 
place in many text-books. It asserts that in remov- 
ing parentheses one should always begin with the 
innermost, proceeding outward. Consider, for exam- 
ple, these solutions : — 


Beginning within Beginning without 

g—latb—(c—d—e)+c]  a—[a+b—(e—d—2) +4] 
=a—[la+b—(c—d+e)+¢] =a—a—b+(c—d—e)—c 
=a—[a+b—c+d—e+c] =a—a—b+c—d—e—c 
=a—a—b+c—d+e-—c =a—a—b+c—d+e-—c 
= —b—d+e == —b—d+e 
It is evident that there are fewer changes of sign 
in the second (4) than in the first (8), and also that 


184 THE TEACHING OF ELEMENTARY MATHEMATICS 


the second and fourth lines in the second could have 
been omitted even by a beginner. The only excuse 
for the first plan is that it affords more exercise; but 
on the same reasoning a child would do well to per- 
form all multiplications by addition. 

The negative number is supposed to be the first 
serious crux for the pupil to bear in his journey 
through algebra. Much has been written as to the 
time for its introduction. Some teachers assert that 
it should find place with the first algebraic concepts. 
Others go to the opposite extreme and teach the 
four fundamental processes with positive integers, and 
then go over them again with the negative number. 
Each teacher, like each text-book, has some peculiar 
hobby, and rides it more or less successfully. As 
has been stated, some make much of the idea that 


b] 


— a should be read ‘negative @” instead of the gen- 
erally recognized “minus a,” hoping thereby to avoid 
the confusion thought to be incident to the two 
senses in which “minus” is used; others (and most 
of the world’s best writers) recognize that this two- 
fold meaning of “minus” has become so generally 
accepted as to render futile any attempt at change. 
The very diversity of view shows how unimportant is 
the question of the time and method of presenting 
the subject, and of the language in question. 

The writer has not been conscious of any great 
difficulty in presenting the matter to classes, and 


ee 


TYPICAL PARTS OF ALGEBRA 185 


after trying the various sequences has for some time 
followed this plan: first teach a working knowledge 
of the alphabet of algebra, through the evaluation of 
simple functions; then awaken the pupil’s interest by 
the introduction of some easy equations, including such 
as 4/2 +2 = 8, Vx +1 =3, etc.; then show the neces- 
sity for a kind of number not commonly met in arith- 
metic, developing the negative number and the zero. 

The explanation cannot be very scientific at first. 
The teacher will depend largely upon graphic illus- 
tration and upon matters familiar to the pupil. The 
symbol for 2° below zero, for 50 years before Christ, 
the symbols for opposite latitudes or longitudes, these 
lead to the general symbol for a number on the other 
side of a zero point from the common (positive) 
numbers. The ingenuity of teacher and pupils then 
comes into play in the way of illustrations; the 
weight of a balloon when empty, when full of gas; 
the capital of a man who, having $5000, loses $3000, 
$5000, $6000; and then the combined weight of a 
10 lb. block and a balloon which pulls upward with 
a force of 20 lb., and the advantage of the expression 
“to lb. and minus 20 Ib.” 

With this introduction the graphic representation of 
positive and negative numbers on a line is a matter 
of no difficulty. After this the more scientific pro- 
cedure, showing the necessity of the negative number 
if we are to solve an equation like x + 3 =1, and the 


186 THE TEACHING OF ELEMENTARY MATHEMATICS 


definition of negative numbers and of absolute values, 
complete with little difficulty the elementary theory. 
It must not be supposed that the negative number 
is necessarily approached by the graphic method. 
This is the more psychological, but not the more 
scientific from an algebraic standpoint. Comte long 
ago pointed this out, and all advanced works on the 
theory now recognize it. ‘‘As to negative numbers, 
which have given rise to so many misplaced discus- 


5) 


sions, as irrational as useless,’ says Comte, “we must 
distinguish between their abstract signification and 
their concrete interpretation, which have been almost 
always confounded up to the present day. Under 
the first point of view, the theory of negative quan- 
tities can be established in a complete manner by a 
single algebraical consideration.”! It is, however, 
impossible to enter into any extensive discussion of 


the theory at this time. 


1 Comte, The Philosophy of Mathematics, translated by Gillespie, N. Y., 
1851, p. 81. 
\ 2 Most teachers have access to Chrystal’s Algebra, or Fine’s Number 


: System of Algebra, and these works give satisfactory discussions of the 


‘subject. For a résumé of the matter from the educational standpoint — 


it is well to read the Considérations générales sur la théorie des quan- 
tités négatives, et objections que on y a opposées, in Dauge’s Cours 
de Méthodologie mathématique, 2. éd. p. 125. But the best works for 
the advanced student are the comparatively recent German treatises by 
Stolz, Baltzer, Biermann, ef a7, or Schubert’s Grundlagen der Arith- 
metik in the Encyklopadie der mathematischen Wissenschaften, 1. Heft, 
Leipzig, 1898. 


ae ee 


TYPICAL PARTS OF ALGEBRA 187 


Of course the teacher will not leave the subject 
without having the pupil understand that the signs 
+ and — have each two distinct uses, one that of 
symbols of operation, as in 10 — 8, the other that of 
quality, as— 8. As Cauchy puts it, “The signs + 
and — modify the quantity before which they are 
placed as the adjective modifies the noun.” Similarly, 
the words plus and minus have (as noted on p. 184) 
two distinct uses, as in ‘‘a plus quantity” and “a 
plus J.” It is true that it has been suggested that the 


) 


expressions “plus a” and “plus quantities” should 
give place to “positive a” and “ positive quantities,” 
these terms being more precise. But much as we may 
theorize upon the desirability of such usage, the fact 
remains that colloquially the shorter expressions are 
generally used by the world’s great mathematicians, 
and will probably continue to be so used. 

The older text-books often contain a great deal of 
worthless matter, and worse, about proving that ‘minus 
a minus is plus,’ and “minus into minus is plus,” etc. 
Of course it is impossible to prove any such thing de 
novo. Mathematicians recognize perfectly well that 
—a:—b=+ab because we define multiplication involv- 
ing negatives so that this shall be true. If we should 
change the definition we might change the result of the 
multiplication. All that is to be expected of the teacher 
is that it should be shown why the mathematical world 
defines — a-— 4 to mean the same as +a: + 4, why any 


188 THE TEACHING OF ELEMENTARY MATHEMATICS 


other definition would be inconsistent. These things 
are easily explained, but the text-book “proofs” of 
the last generation have now been discarded. The 
favorite one of these “proofs”’ was this: Since multi- 
plying — 0 by a gives — ad, therefore if the sign of 
the multiplier is changed, of course the sign of the 
product must also be changed. As a proof, it is like 
saying that if A, a white man, wears black shoes, i 
therefore it follows that B, a black man, must wear : 
shoes of an opposite color. 

Checks— When a large transatlantic steamer not 
long since ran upon the rocks near Southampton, 
the captain announced that he had made an error of 
a few miles in his calculations. Thousands and 
thousands of dollars lost, hundreds of lives jeopard- 
ized, just because a simple calculation had not been 
checked! And yet one of the first things that every 
computer learns is the necessity for checking each 
operation, a necessity which should be impressed 
upon the student of algebra from the first day of his 
course. It is a matter of no moment whether we say 
“check” or “ prove” or “verify”; mathematicians 
| probably use the first most often; but it is a matter of 
| greatest moment that we see that each step is right. 

What checks the teacher shall require depends 
somewhat upon the pupils. A few of the more com- 
mon ones will be suggested, it being understood that 


the list is not exhaustive. 
ey 


TYPICAL PARTS OF ALGEBRA 189 


In solving an equation the one and only complete 
check is that of substituting the result in the original 
equation (in the statement of the problem if there be 
one). It makes no matter what axioms we use or 
how carefully we proceed; a result is right if it 
“checks,” and wrong if it does not. As Professor 
Chrystal says: “The ultimate test of every solution is 
that the values which it assigns to the variables shall 
satisfy the equations when substituted therein. No 
matter how elaborate or ingenious the process by 
which the solution has been obtained, if it do not 
stand this test, it is no solution; and, on the other 
hand, no matter how simply obtained, provided it 
foo stands this test, it is a solution.’ 1: Professor 
Henrici expresses the same thought in another way: 
“Simplifications of equations follow in senseless mo- 
notony, until the poor fellow really thinks that solv- 
ing a simple equation does not mean the finding of a 
certain number which satisfies the equation, but the 
going mechanically through a certain regular process 
which at the end yields some number. The connec- 
tion of that number with the original equation remains 
to his mind somewhat doubtful.” ? 

To illustrate, consider the equation *++2=3. Sup- 
pose we multiply these equals by +—2, the results must 
ae equal, and 27—4=3%—6, whence 2#°—34+2=0. 


1 Algebra, Vol. I, p. 286. 
2 Presidential address, Section A, British Assn., 1883. 


190 THE TEACHING OF ELEMENTARY MATHEMATICS 


Solving, +=2 or 1. But although we have followed 
axioms strictly, s=2 will not satisfy the original equa- 
tion. So with any equation, the pupil who checks his 
work is master of the situation; answer books are only 
in the way, save in the case of unusually complicated 
results, and the pupil knows as well as the teacher (per- 
haps better) whether his result is right or wrong. “A 
habit of constant verification cannot be too soon encour- 
aged, and the earlier it is acquired the more swiftly and 
almost automatically it is practised.” 4 

A very useful check, applicable to the operations of 
algebra, is that of arbitrary values. Whatever values 
are assigned to a and 4, (a + 4} must always equal a? + 
2ab+ 6%. In other words, we may substitute arbitrarily 
any values for a and 4, and see if the two forms agree. 
E. g., let a= 2, b= 3; then (24+ 3h = 274+2-2+3-+4 34 
which is true because each is 25. Or suppose a pupil 
asserts that (47+ 34—5)(?+24-—1)=2445234+7 
— 13%+5; 1s the result correct? Substitute any arbi- 
trary value for x, say 1, and the question reduces to this, 
Does —1:2= —1? Since it does not, there is evidently 
an error. The arbitrary value 1 is usually a good one 
unless zero enters somewhere; it does not check the 
exponents, since any power of I is I, but mistakes 
are not usually made there. Of course in checking 
a case like (#8—1)/(*#—-1)=2#7+2%+4+1, it will not 


1 Heppel, G., Algebra in Schools, the Mathematical Gazette, February, 


189s. 


TYPICAL PARTS OF ALGEBRA IQ! 


do to use the value 1 for x; and in general those 
values should be avoided which make any expression 
zero. 

Another check extensively used by mathematicians is 
that of homogeneity. The name is long, but the check 
is simple. ‘“ At present, although ‘homogeneous’ is 
usually defined somewhere in the first three pages of 
a school algebra, the school-boy never knows anything 
about its meaning, as he has not been used to apply it.” 
The check simply recognizes the fact that if two inte- 
gral functions are homogeneous, their sum, difference, 
product, and powers, are homogeneous. /.g., the prod- 
uct of a2 + ae? and a®+ ad may be a? + a9? 4+ att + 
a*}?, because the product of a homogeneous function of 
the third degree and one of the second must be one of 
the fifth ; but if the result is given as a° + a°J7 + a®b + 
av? there must be an error, because the result is not 
homogeneous. Since homogeneous functions play such 
an important part in mathematics, this check is of more 
value than at first appears. 

Still another check, less extensively used, but so 
easily applied as to be valuable, is that of symmetry. 
If two functions are symmetric with respect to cer- 
tain letters, their product, for example, must be sym- 
‘metric with respect to those letters. .¢., 2*-ay+y)’ 
and #2+ +4y+ 7% are symmetric with respect to 4 and 
'y, since these may change places without changing 


1 Heppel, G., in the Mathematical Gazette, February, 1895. 


192 THE TEACHING OF ELEMENTARY MATHEMATICS 


the forms of the functions. Hence a*+ 77)?+ 44 
may be their product, but not #4—x°y+427y?+273+ 74, 
although it checks as to homogeneity and for the 
arbitrary values, r= 1, y= 1. 

The first two of the checks mentioned should be 
in constant use by the student; the others are valu- 
able, but not indispensable. 

Factoring has already been mentioned as a subject 
of supreme importance in algebra. Pupils waste 
much time in performing unnecessary multiplications 
and in not resorting more often to simple factored 
forms. For example, the student who begins the 
solution of the equation 


243+327-4r-I 
r—1 


=2?4+4r—-I1, 


by clearing of fractions, gets into trouble both theo- 
retically and practically; he introduces a root which 
does not belong to the equation, and he causes him- 
self some unnecessary work. He should see at a 
glance that r—1 is a factor of 24%3+327-—4%—-1, 
and can easily do so if he understands the elements 
of the subject. : 

While it must be admitted that the recent text- 
books have improved upon the older ones in the 
matter of factoring, there is room for further im prove- 
ment. The subject is often divided into “cases,” 
_often with almost no difference, as with 27+ ar+ 8, 


TYPICAL PARTS OF ALGEBRA 193 


v—ax+b, #+ar—4, etc., thus leading to a style 
of treatment that is depressing. It is true that the 
arrangement of a page of exercises like 27+ ar + 4, 
followed by another of the type +? — ar + 4, etc., has 
educational value, but it is also true that the arrange- 
ment is not a good one. It reminds one of the six- 
teenth century plan of having one rule for the quad- 
ratic #7+fr+g=0, another for 7*—fr+g=0, another 
for #7+ %x=9, and so on. The favorite answer to 
all this is that pupils cannot generalize and take the 
single type 27+ ar+ 0, where a or 4 may be either 
positive or negative; but the experience of the best 
teachers shows that pupils can generalize much earlier 
than some of our text-books would seem to indicate. 
Some special forms must always precede the general; 
but to give only special forms, never referring to the 
general type, is a serious error. 

The fact is, there are only a few distinct types of 
factored expressions that are of much value in subse- 
quent work. The most important are (1) ab + ac, the 
type involving a monomial factor; (2) aa*+ér-+<c, the 
general trinomial quadratic in x, (3) cases involving 
binomial factors of the form x—a. Of course for 
the beginner these must be still further differenti- 
ated; but problems not involving these three cases, 
such as the factoring of 


a*+ xy? + 74, and 272+ 73+ 232 — 3 xyz, 
A | 


194 THE TEACHING OF ELEMENTARY MATHEMATICS 


have value rather as mental gymnastics than as cases 
to be used in subsequent work. 

The type a#*+6r+¢ includes certain special cases 
which must be considered briefly before the general 
one, such as 2#7+2ar+a%, 227—-a*, 244+(a+6)r+ab, 
in which a and J may be either positive or negative. 
These special cases are satisfactorily discussed in 
most text-books. The general type, az7+ dr+ 6, is | 
not, however, so well treated. There are numerous 
methods of attacking it, but only two are valuable 
enough for mention here. The first will be under- 
stood from the following: 


62+1747+12=627+9044+8%4+4+12 
=34(24+ 3)+4(244 3) 
=(34+4)(24 + 3). 


That is, the 17 is separated into two parts whose 
product is 6:12, and the rest of the work is simple. 
In general, in az*+dxr+¢, the J is separated into 
two parts whose product is ac. The reason for this 
is easily seen by considering that 


(mx + 2) (mx + n')= mu! x* + (mn! + m'n)x + un; 


that is, that the coefficient of + is made up of two 
parts, mz’ and m!'nx, whose product is m' + nn. 

The other plan consists in making the coefficient 
of #* a square, thus: 


6229+ 1744+ 12=3(362°+17-6%+4 72). 


TYPICAL PARTS OF ALGEBRA 195 


Now let z=6-4, and we have 


4(2+172+72)=34(¢4+9)(¢ 4+ 8) 
=4(64+9)(6%+ 8) 
=(24+3)(34+4). 


Which of these plans is followed is immaterial, the 
rationale of each being easily explained. But it is 
needless to say that the cut-and- try method often given, 
of taking all possible factors of 6 #7 and of 12 and guess- 
ing at the proper combination, has little to recommend it. 

The cases involving binomial factors of the form 
4—a are perhaps the most important of any which 
the pupil meets in his elementary work, since they 
enter so extensively into the theory of equations. 
They are best treated by the remainder theorem, 
which has long found place in the closing pages of ” 
many advanced algebras, where it could not be used 
to any extent. The theorem asserts that the remain- 
der arising from dividing an integral function of + 
by x —a can be found in advance by putting a@ for 
# in the given function. A#.g., in dividing 2+— #3 
+52*—16%+11 by x—1 we know that there will 
be no remainder, for 1—1+5—16+11=0; but if 
#%—2 is the divisor, there will be a remainder 7, for 


2*—2°+5-2?—-16-2+11=16—8+20—32+11=7. 


Similarly, x’—y" is divisible by x—y, for if y is 
put for x, 2” —y" = y" —y"=0; but it is not divisi- 


196 THE TEACHING OF ELEMENTARY MATHEMATICS 


ble by «+ y, ze, by x—(—y), for if —y is put for 4, 
(—yyt — yy" = — yy" — y = —2y", the remainder. The 
theorem is easily proved, and its usefulness in ele- 
mentary algebra can hardly be overestimated. The 
proof, condensed more than advisable for beginners, 
is as follows: 

Let f(x) be the dividend, x—a the divisor, g the 
quotient, 7 the remainder. 

Then f(*4)=(*—4@)¢+~7, in which ~ cannot con- 
tain x. 

This being an identity is true for all values of +, 
and hence for += a. 

But if a is put for x, we have f(a)=~. 

Z.e., the remainder is the same as f(r) with @ put 
tor iA. 

A teacher will have no difficulty in putting this 
into a form easily comprehended by beginners. The 
theory is not difficult, and the practice is very 
simple. 

It is unfortunate that, having spent considerable 
time upon the subject of factoring, so many text- 
books thereupon relegate it to the mathematical 
garret. The next chapter is usually upon highest 
common factor, in which the pupil is led to make 
as little use of factoring as’ possible! After con- 
sidering the lowest common multiple, the text-books — 
next proceed to fractions, and here the pupil is 
led to use the highest common factor in his_reduc- 


V 


TYPICAL PARTS OF ALGEBRA 197 


tions, which we rarely do in practice, but other- 
wise the important subject of factoring sinks into 
disuse. 

What is the remedy for this evil? The answer 
appears when we consider the common uses to which 
the mathematician puts the subject. He has two 
uses for it, the first being in the solution of equa- 
tions, and the second in shortening his work, as in 
the reduction of fractions to forms more easily han- 
dled. Hence it is proper to follow factoring at once 
with some simple work in equations, and as soon as » 
fractions are met to use factoring in all simple re- ° 
ductions, reserving the highest common factor for 
cases of real difficulty. 

The application of factoring to the solution of 
equations is very simple, if the pupil knows what 
it means to solve an equation like 


a" +ax® +--+ +n=0, 


‘namely, to find a value of # which shall make the 
first member zero. That is, the value of zx which 
makes +—a=o0 is evidently a. The values which 
make 22—3x—4=0, or, what is the same thing, 
(2—4)(4+1)=0, are evidently 4 and — 1, because 
mer 4 we have 0o-5=—0, and if +=—1 we have 
—5:0=0. Similarly, the values which make 


#— ax=0, or (4 +2)(*#—2@2)=0, 


198 THE TEACHING OF ELEMENTARY MATHEMATICS 


are evidently 0, —a, +a. In this way a consider- 
able number of equations with commensurable roots 
should be given, together with problems involving 
equations of- degree above the first, thus at the same 
time adding to the interest in the subject, giving drill 
in factoring, and laying a rational foundation for 
quadratics. A pupil so trained would not, on reach- 
ing the chapter on quadratics, waste his time ‘‘com- 
pleting the square” in the solution of such equations 
as a2@+2%7=0, or z47+57++6=0. It takes but little 
time to introduce this work, whatever text-book is in 
use, and the benefit derived is evident. 

In the treatment of fractions, to apply the Eucli- 
dean method of highest common factor! to the reduc- 
tion of forms like 

2+ 7xr-+ 10 a 24+ 627+ 34—I10 

x? +90x4+ 14 4824+ 5x%—-14 
is to encourage the pupil to waste time and to forget 
his elementary work in factoring. 

The quadratic equation, often looked upon as the 
final chapter of elementary algebra, seems peculiarly 
open to mechanical treatment. Add the square of half 
the coefficient of +, extract the square root, transpose — 
this is the rule; the validity of the result is not consid- 

1“ Then there are processes, like the finding of the G.C.M., which 
most boys never have any opportunity of using, excepting perhaps in the 


examination room.” Henrici, Presidential address, British Association, 
Section A, 1883. 


TYPICAL PARTS OF ALGEBRA 199 


ered essential. The reason for this procedure is doubt- 
less historical; the early mathematicians were forced to 
solve in this way, and the tradition has endured to 
the present. 

But if we are to follow this mechanical route, we may 
well go even farther. For practical purposes the pupil 
eventually needs to be able to write down at sight the 
roots of equations like 27+ 2%+3=0, without stop- 
ping to “complete the square’; for this purpose the 
formula 


ane £2 VP—49 


should be as familiar to him as the multiplication table. 
To use the method of the completion of the square in a 
thoughtless way with every equation has neither a cul- 
ture value (since the logic is concealed) nor a utilitarian 
value (since it is an unnecessarily tedious way of reach- 
ing the result). 
| The best plan of attacking the quadratic equation is, 
as already intimated, through factoring. The plan is 
simple, it is general (not being limited to quadratics), 
it can be introduced with factoring and continually 
reviewed until the chapter on quadratics is reached, and 
at the same time it keeps the subject of factoring fresh 
in mind. When the chapter on quadratics is reached, 
the student is already able to handle the ordinary run 
of manufactured problems, those which ‘come out 
even ”’— with small integers for roots. Those involving 


200 THE TEACHING OF ELEMENTARY MATHEMATICS 


large numbers, however, require other methods, and 
this leads to the completion of the square, an expression 
derived from the old geometric method of. solving the 
quadratic equation. The outcome of this method should 
be the proof of the fact that if 


elt peg = Ot £4 VP 40, 


or, if preferred, the formula for solving ax*.+ dr +c¢=0. 
This formula, logically developed, is so important as to 
demand sufficient application to fix it in mind for use in 
the subsequent parts of algebra. That a pupil should 
“complete the square” every time he runs against an 
equation like 14 + + + 1 =0 is as senseless as to require 
him to add three 13’s when he wishes the product of 3 
and 13. 

Some text-books give one or two other methods of 
solving the quadratic, but these serve to confuse rather 
than assist the pupil. Their interest is more historical 
than educational. That the teacher may see that the 
standard solution is not the only one, however, a few 
historical devices may be of service: 

Method of Brahmagupta (b. 598) and Bhaskara 
(b. 1114). 

Given avi+ br=c. 


Then 4a%x? + 4abx = Aac, 


1 Matthiessen, Grundziige der antiken und modernen Algebra, 2. Ausg., 
Leipzig, 1896, p. 282. 


TYPICAL PARTS OF ALGEBRA 201 


407+ 4abxr + P=4ac+ &, 
ah 2ax+b=4VAac+ &, 


a ee bv ies + 6°), 
2a 


This plan, here given in complete form with modern 
symbols, is sometimes called the Hindu method. It has 
the advantage of avoiding fractions until the last step. 

Method of Mohammed ben Musa (about 800, see 
p. 151) and Omar Khayyam (d. 1123, the author of 
the Rubaiyat), one of several given by them, and 
based on geometric considerations.} 


Given ee. 
Benen | \*.: Ppt = (+-£), 
oe g+£=(x-2), 
and A eee 
or rata vo+e 


This plan is essentially the one now in general use. 
Method of Vieta (1615).? 


Given v+taxr+b=0. 
Let H=u+¢. 
Then #+(22+a@)u+(2+az+ 6)=0. 


1 Matthiessen, p. 309. 2 Matthiessen, p. 311. 


202 THE TEACHING OF ELEMENTARY MATHEMATICS 


Since but one condition has been placed upon zw + g, 
we may impose another, and let 


2Z2+a=0, 
whence sas 
2 
and uw? —1(a*— 4b)=0, 
whence x=ut+z=—lathva—4e. 


Here there has been no “completion of the square.” 
Method of Grunert (1863).} 


Given x?+ar+b=0. 

Let L=ut+s. 

But (a#+2f—2u(u+2)+(e—2)=0. 
. a=— 2u, and d= 22 — 24. 


w= —% and z=+1v2a?— 44, 


from which ~ is easily found. 
Fischer’s trigonometric method (1856) is one of sev- 
eral of this class. 


Given x* — pxr+q=0, with 22> 44. 
ee #, = 2 -.COS* @, one root, 
and #5. p.- Sin’ , the other. 
Then +7, +%7,=p(cos*¢ + sin? ¢) = 4, 
and. 41%, =p" (sing: cos PY = } p?- sin?2 ¢. 
But 4% = 9, .. sin 2p = 2G. 


1 Grunert’s Archiv, Bd. 40. 


TYPICAL PARTS OF ALGEBRA 203 


For example, to solve 77 — 93.7062 + + 1984.74 =0. 


Here 26= 71° 57' 44."6, .. b = 35° 58! 52.'3, 
whence #1 = 61.3607, % = 32.3454. 


The problem shows that trigonometry is able materially 
to assist in the solution of certain kinds of quadratic 
equations. 

There are many other devices for solving the quad- 
ratic, for which the reader must, however, be referred 
to the great compendium of Matthiessen. Enough of 
these plans have been suggested to show that a de- 
parture from the single one in general use, for the 
purpose of emphasizing the method of factoring and 
the use of the formula, is not a novelty to be feared; 
it is merely to make a judicious selection from the 
abundance of material at hand. 

Equivalent equations — To the student who has not 
been taught that there is no escape from the check- 
ing of the roots of an equation, and that extraneous 
roots are liable to enter with any one of several com- 
mon operations, it seems sufficient to blindly follow 
the axioms until a solution is reached. But this 
is so far from the case, and the text-books offer so 
little upon the subject, that a brief statement con- 
cerning the matter may be of service to teachers. 

While it is true that the solutions of equations de- 
pend upon a few well-known axioms, these axioms 
may lead the student into difficulty. For example: 


204 THE TEACHING OF ELEMENTARY MATHEMATICS 


Let =A. 


Then,tmultiplying by 7 29 ae 


Subtracting a3, #-—A@=ar— a’ 
Factoring, (4 +a)(*4—a)=a(4— a). 
But z= a, “. 2a(%—a@)=a(e—a@). 


Dividing by + — a, 
20d Orne 11 


Here every step follows from the preceding one by 
the application of a common axiom, and yet the 
result is absurd. 

Pupils are apt to place undue weight upon demon- 
strations apparently valid but in reality fallacious. 
But as J. Bertrand, the French algebraist, says, ‘ Com- 
mon sense never loses its rights; to set up against 
evidence a demonstrated formula is about like telling 
a man that he is dead because you happen to have 
a physician’s certificate to that effect.” 

This tendency of pupils and this testimony of M. 
Bertrand suggest the question: What limitations are 
there on the use of the axioms? To answer this 
question requires the definition of eguzvalent equa- 
tzons. Two equations are said to be equivalent when 
all of the roots of either are roots of the other. 
E.g,*#+3=3"—-1 and r+1=3(4— 1) are equiva- 
lent equations, for x= 2 is a root, and the only root, 
of each. But: += 3) ando7* 6 tareynot equivalenm 


TYPICAL PARTS OF ALGEBRA 205 


for r=— 3 is one root of the second, but it is not a 
root of the first. 

It is axiomatic that if equals are added to equals 
the results are equal, but it does not follow that the 
resulting equation is equivalent to either of the orig- 
inal ones. £.g.: 


If SoS 

then Pog tees ty 
Adding, re, 
Solving, 4=1o0r—2. 


The —2 is a root of #7+2x=2, but not of x=1 
nor of #7=1. The equation #2+2=2 is not equiva- 
lent to either of the others. 

It is also an axiom that if equals are multiplied by 
equals the results are equal. But it does not follow 
that the resulting equation is equivalent to the others. 
Eig. if «—1=1, and we multiply by ++1, while 
it is true that 27—1=%+1, it does not follow that 
its roots are the same as that of r—1=1. They 
are not, for 27—1=2z+1 has fwo roots, 2 and —1, 
but —1I is not a root of the first equation. And in 
general, if we multiply by a function of x we intro- 
duce (if the equation is integral) one or more new 
roots, “extraneous roots” as they are called. 

Similarly, if += 5, then 27 = 25, 2° = 125, 2* = 625,--; 
but the second equation has one root which the first 


206 THE TEACHING OF ELEMENTARY MATHEMATICS 


has not, —5; the third has two which the first has 
not, 5(—4+4V-— 3); the fourth has three extra- 
neous to the first, and so on. 

Furthermore, the axiom of dividing equals by 
equals needs watching. If 42+ 22%—-2z=0, then by 
aividind: yiby °4,) 47-2 4 1 10, ole +V 2. 
These are roots of the original equation, but they 
are not the only ones; +=0 is also a root. And in 
general, dividing by a function of x loses one or 
more roots. 

In dealing with radical equations the difficulty is 
even more pronounced. When we deal with radicals 
it is customary to consider only the sign expressed 
before them, or if none is expressed to understand 
the plus sign. That is, we consider the value of 
V4+ V9 to be 2+3=5, and not (+2)+(+ 2) Ean 
— I, I, or —5. This is purely conventional; it has 
simply been agreed that in elementary work the stu- 
dent shall not be bothered with the + unless it is 
expressed, as t+V44+V9Q=5, —1, 1,°—5. Since the 
square root of 4 is either 2 or — 2, it is evident that 
the plan is not very scientific; but so long as it is 
understood no great harm can come from it. So when 
we are dealing with the radical equation V x — 1 = 3, we 
seek the root which satisfies the equation +Vx—1 
= 3 and not —-Vx—1=3, although of course the 
square root of +—1 is both plus and minus. With. 
this understanding, consider the following solution: 


TYPICAL PARTS OF ALGEBRA 207 


Given VE— 8 =i — Ve—2, 
Squaring, 4#-5=1+4—-—2-—-2Vr-2., 


Hence BNA A ima A, » 


oA 
i 


and = 6. 


But on substituting 6 for 7, we have 


Vi=1—V4, 
or I=1I—2. 


That is, if we understand Vi to mean the positive 
Square root of 1 and not the negative one, 6 is not 
a root. It is therefore called extraneous, and the 
equation is said to be insoluble. However unscien- 
tific this may seem, the limitation of the sign before 
the radicals in such way as to make many equations 
insoluble, it has high mathematical sanction! At any 
rate, it is evident that the application of the axioms 
gives rise to roots commonly considered extraneous. 
Considerationg such as these show how necessary 
it is to make more of the logic of algebra than is 
usually done. The average pupil in algebra seems 
quite content if able to say, “‘Transposing I got this, 
and by squaring I got this, and the next step came 


1 Following Chrystal, Todhunter, Hall and Knight, and the majority 
of writers, Va should be considered a quantity having one and not two 
values, although the algebra of C. Smith and the article by Professor 
Kelland in the Encyclopedia Britannica make Va have two values.” 
G. Heppel in the Mathematical Gazette, February, 1895. 


208 THE TEACHING OF ELEMENTARY MATHEMATICS 


from dividing,” etc., with no thought as to the legiti- 
macy of the process. He gives with each step the 
“How,” and teachers are often content; but this is 
of relatively minor importance, the great questions to 
be asked at each step being, “ Why is this true?” 
and, “Is the process reversible?” 

Simultaneous equations and pranhesihere is often 
an objection raised against the introduction of graphs 
in elementary algebra, that there is no reason for © 
thus anticipating analytic geometry. We are told 
that algebra and geometry are separate sciences, 
although this separation is really a recent event in 
the history of the two subjects. What a striking 
little rebuke to those who would build impassable 
barriers between the branches of mathematics which 
we vainly try to separate by distinctive names, is 
the epigram of Sophie Germain, “ Algebra is only 
written geometry — geometry merely pictured alge- 
bra!” +! The introduction of the graph is so simple, 
and throws such a flood of light upon simultaneous 
equations, that teachers who have used the plan 
rarely abandon it. A pupil can understand much 
more fully why two linear equations with two un- 
knowns are in general simultaneous, if the matter is 
brought to the eye, by the two lines which represent 


1 Lalgébre n’est qu’une géométrie écrite, la géométrie n’est qu’une 
algébre figurée. It recalls Goethe’s description of architecture as “ frozen 
music,” eine erstarrte Musik, which struck Mme. de Staél as so felicitous. 


TYPICAL PARTS OF ALGEBRA 209 


these equations, than he can by an analytic proof. 
He sees, too, why the attempt to solve the set 


24+Oy=5, 34 +97=7, 


fails. If he is told that in general three linear equa- 
tions are not simultaneous, the reason is more clear 
when supplemented by the pictures, the graphs, of 
the equations. When he finds that in spite of the 
general fact just stated, the special equations 


2 H3 P= 0, 1224-849 = 10,7 — 72 


are simultaneous, and that, indeed, others can be 
added to the set, as 


47x#+13y = 26, 154+ 15 y = 30, etc., 


the mystery of the matter vanishes as soon as the 
gr’ phs are plotted. 

‘Similarly for an equation of the second degree com- 
bined with another of that degree or with a linear 
equation. While there is a simple proof that in gen- 
eral two simultaneous quadratics cannot be solved 
without recourse to a quartic equation, most students 
fail to appreciate the fact until they have the assist- 
ance of graphs. Most pupils who have “finished” 
quadratics would expect to be able to solve the set 


+ 3ay+477+5¢4+67+7=0, 


+227 —j7"— 134 —-—17y — 20=0, 
P 


210 THE TEACHING OF ELEMENTARY MATHEMATICS 


and would wonder at their inability to handle it. 
They cannot understand why such an innocent look- 
ing set as 
Mtry=7,¢+7= 11, 

(partly soluble by quadratics if one makes the lucky 
hit) should give them trouble. They are satisfied 
with one or two roots of the set 

4+ 2ry +972 -7=0, 274+ 3247+ 277-8 =0, 
or with half a dozen, if by the introduction of extra- 
neous ones they can get together that number. ‘The 
curious thing is that many examination candidates 
who show great facility in reducing exceptional equa- 
tions to quadratics appear not to have the remotest 
idea beforehand of the number of solutions to be 
expected! and that they will very often produce for 
you by some fallacious mechanical process a solution 
which is none at all.’ 

A valuable exercise for a class which has devoted 
a little time to graphs, is to consider the graphic sig- 
nificance of each new equation obtained in the solu- 
tion of a pair of simultaneous equations involving two 
unknowns. Each equation properly derived must rep- 
resent a graph passing through the intersections of 
the graphs corresponding to the first two. £.g.: 


I. Given x + 7 =O, 
2. and r+y =3. 
1 Chrystal, Presidential address, 1885. 


TYPICAL PARTS OF ALGEBRA 211 
3. Then 22 — xy +7" = 3, by division. 
4. A+2xy+7"=9, by squaring (2). 
5. “. ry = 2, by subtracting and dividing. 


Equation (3) represents an ellipse which passes 
through all the intersections of the graphs (1) and 
(2) except the point at infinity; (4) represents two 
parallel lines, only one of which passes through the 
intersections of (1) and (2); (5) is an hyperbola pass- 
ing through the intersections of the ellipse (3) and 
the parallels (4). The solution then passes on to the 
intersection of the straight line (2) with the par- 
allels (4 —y)? = 1. 

In general, the question of the number of roots 
to be expected, the entrance of complex roots in 
pairs, the conditions rendering equations simultane- 
ous, or inconsistent, or impossible, these necessary 
and not particularly difficult bits of theory are made 
to stand out much more clearly by the use of the 
graph. 

Methods of elimination — Elementary text-books al- 
ways distinguish several cases of elimination with 
respect to linear equations. These are, (1) addition, 
(2) subtraction, (3) comparison, (4) substitution, and 
possibly (5) Bézout’s method. If those who love 
novelty only knew it, there are numerous other 


1 A problem used by Professor Beman in his teachers’ course in algebra. 


212 THE TEACHING OF ELEMENTARY MATHEMATICS 


methods which might be brought in to give this turn 
to the subject.? 

But for the practical purposes of a beginner there 
are only two distinct methods of much value, (1) 
that of addition, under which subtraction is merely a 
special case, because the sign of the proper multiplier 
to be employed will always reduce the process to 
one of addition; (2) that of substitution, under which 
comparison is merely a special case, for in equating 
x=y—2 and +=3y+4, we substitute the value of 
4 just as much as we compare values. Hence in 
teaching the subject, it is to these two methods 
that especial attention is to be given, the other plans 
suggested by the text-book being shown to be special 
cases. Indeed, before the pupil leaves the subject it 
might not be going too far to show that the method 
of substitution is a special case of the one general 
method of addition. 

Of equal importance with the existence of the two 
methods mentioned, is the question as to their use. 
The pupil will easily find for himself, if permitted to 
do so, that the addition method is usually preferable, 
the other being the easier only in special cases, as 
in that of unit coefficients, or in finding one of two 
values after the other has been ascertained. 

When both equations are of the second degree, the 
student should early be led to see that in general no 


1 See Matthiessen, for example, 


TYPICAL PARTS OF ALGEBRA 213 


solution is possible by quadratics, and that the only 
cases which he can handle with any certainty are 
those involving homogeneous or symmetric functions. 
The methods of attacking these cases are well known 
and need not be discussed here. But in the case of 
symmetric equations it should be noted that most 
text-books lose sight of one of the essential features. 
By the very nature of symmetry the roots must be 
_ identical. Consider, for example, the set 


V4+3ay7+7?-41=0, P+72?4+4+y7—-32=0. 


By the usual method x is found to be 3(—19+ V 329), 
5, or I, four results, as should have been anticipated. 
It therefore follows, without substituting or applying 
any special devices, that y has identically the same 
values because of the symmetry of each function as 
to # and y. Of course the particular value of y to 
be taken with a given value of x is not yet deter- 
mined, but this is usually seen at once by looking 
at the two equations. The failure to recognize all 
this results in serious loss of time; the student gets 
exercise, it is true, but he might more profitably get 
it by solving another set of equations than by failing 
to appreciate one of the essential parts of the theory. 

Complex numbers— As already stated, it is only 
since Gauss, in 1832, brought before the mathemat- 
ical world at large the theory which Wessel and 
Argand had developed, that the complex number has 


214 THE TEACHING OF ELEMENTARY MATHEMATICS 


been well understood. Even now it is only slowly 
finding its way into elementary text-books, such works 
usually saying (of course ‘‘between the lines’’), “ Here 
is ~—1, and we do not know what it means or what 
to do with it, and we will hasten over it with as 
little trouble as possible.” Where in the course in 
algebra this perfunctory treatment shall be given has 
been the subject of not a little discussion, as if it 
made any difference. If the student is to receive 
nothing, what matter whether that nothing comes 
this month or next? 

What, then, should be done with the subject? 
When should it be introduced, and how should it be 
explained ? | 

It is an educational maxim already several times 
invoked in these pages, that a subject is best intro- 
duced just before it is to be used. As soon as we 
reach quadratic equations as a distinct subject we 
meet complex numbers. Equations like 22+ 1=0, 
a*+24+5=0, and in general 77+ pr+¢7=0 where 
?'°<49, give rise to roots involving imaginaries. 
Hence it follows that the chapter on complex num- 
bers logically precedes that on quadratic equations. 
Whether it psychologically precedes depends upon its 
difficulty. 

The difficulty of the chapter has been overrated 
because it is only recently that teachers as a class 
have known anything about the subject. In reality 


TYPICAL PARTS OF ALGEBRA 215 


the graphic treatment of the complex number is no 
more difficult for the pupil who is ready to begin 
quadratics than is that of the negative number to 
the one about to take up the theory of subtraction. 
Teachers are therefore urged, even at the expense of 
a week’s work outside the text-book—if that be a 
hardship —to present the elements of this graphic 
treatment.} 

The applied problems of algebra are usually 
even more objectionable than those of arithmetic. 
When the science began to find place in the schools 
there had accumulated a large number of examples 
which by arithmetic were puzzles, but by algebra 
offered little difficulty. These were incorporated in 
the new science, and they have remained there by the 
usual influence of two powerful agents—the conserv- 
atism of teachers and the various kinds of state ex- 
aminations. To this latter influence is to be charged 
the greatest amount of blame in the matter, not as 
to the individuals who set the examinations, but to 
the inherent evil (possibly a necessary one) of the 
system. Certain of the best teachers of a country 
know that time is wasted over some particular line 
of problems; they would like to omit them, but their 


1 One of the best elementary presentations of the subject is given in 
Fine’s Number System of Algebra, Boston, 1890, a book which should be 
upon the shelves of every teacher of this subject. For a classroom treat- 
ment, see Beman and Smith’s Algebra, Boston, 1900. 


216 THE TEACHING OF ELEMENTARY MATHEMATICS 


hands are tied by the necessity that their pupils shall 
pass a certain examination (civil service, college — 
for these are often among the most objectionable, 
regents’, teachers’, etc.). On the other hand, many 
of the examiners are among the most progressive 
educators. They too would like to see the mathe- 
matical field weeded and conservatively sown anew. 
But their hands are also tied by the system. As a 
progressive English examiner once remarked to the 
writer, “I know that this problem should have no 
place in the examination, but I cannot replace it by 
a modern one because the schools are not up to such 
a change; their text-books do not prepare for it.” 
Speaking of this effect of the examination, Pro- 
fessor Chrystal has not hesitated to express his views 
with perfect frankness. ‘The history of this matter 
of problems, as they are called, illustrates in a singu- 
larly instructive way the weak point of our English 
system of education. They originated, I fancy, in the 
Cambridge Mathematical Tripos Examination, as a 
reaction against the abuses of cramming book-work, 
and they have spread into almost every branch of 
science teaching — witness test-tubing in chemistry. 
At first they may have been a good thing; at all 
events the tradition at Cambridge was strong in my 
day that he who could work the most problems in 
three or two and a half hours was the ablest man, 
and, be he ever so ignorant of his subject in its 


TYPICAL PARTS OF ALGEBRA 217 


width and breadth, could afford to despise those less 
gifted with this particular kind of superficial sharp- 
ness. But, in the end, it all came to the same: we 
prepared for problem-working in exactly the same 
way as for book-work. We were directed to work 
through old problem papers, and study the style and 
peculiarities of the day and of the examiner. The 
day and the examiner had, in truth, much to do with 
it, and fashion reigned in problems as in everything 
else.”1 Still more pointedly he says: “All men 
practically engaged in teaching, who have learned 
enough, in spite of the defects of their own early 
training, to enable them to take a broad view of the 
matter, are agreed as to the canker which turns 
everything that is good in our educational practice 
to evil. It is the absurd prominence of written com- 
petitive examinations that works all this mischief. 
The end of all education nowadays is to fit the 
pupil to be examined; the end of every examination 
not to be an educational instrument, but to be an 
examination which a creditable number of men, how- 
ever badly taught, shall pass. We reap, but we omit 
to sow. Consequently our examinations, to be what 
is called fair—that is, beyond criticism in the news- 
papers— must contain nothing that is not to be found 
in the most miserable text-book that any one can 
cite bearing on the subject.... The result of all 


1 Presidential address, British Association, Section A, 1885. 


218 THE TEACHING OF ELEMENTARY MATHEMATICS 


this is that science, in the hands of specialists, soars 
higher and higher into the light of day, while edu- 
cators and the educated are left more and more to 
wander in primeval darkness.” ! 

This evil, which we have not yet the ingenuity to 
avoid, stares the teacher in the face when he would 
replace obsolete matter by problems which have the 
stamp of the generation in which we live. It is not 
that these problems about the pipes filling the cistern, 
the hound chasing the hare, the age of Demochares, 
and the number of nails in the horse’s shoe, are not 
good wit-sharpeners, and possessed of a kind of in- 
terest; but we have now a large number of equally 
good wit-sharpeners possessed of a living interest, 
problems relating to the life we now live, and to the 
simple science the pupil is now studying. “I some- 
times feel a doubt, however,’ says a recent writer, 
“whether boys really enjoy being introduced to such 
exercises as ‘A says to B, how much money have 
you got?’ and B makes a very singular hypothetical 
reply; or to the fish whose body is half as long 
again as his head and tail together, while head and 
tail have given relations of magnitude. I cannot but 
suspect that there is something unpractical in these 
problems.” These historical problems have some 
value as history and some interest from their very 


1 Presidential address, 1885. 
2 Heppel, G., in the Mathematical Gazette, February, 1895. 


TYPICAL PARTS OF ALGEBRA 219 


absurdity, but it is to be hoped that the rising gene- 
ration of teachers may see them laid aside. ‘A more 
rational treatment of the subject, introducing from 
the beginning reasoning rather than calculation, and 
applying the results obtained to various problems 
taken from all parts of science, as well as from 
everyday life, would be more interesting to the stu- 
dent, give him really useful knowledge, and would be 
at the same time of true educational value.” } 

It is a serious question whether America, following | 
England’s lead, has not gone into problem-solving 
altogether too extensively. Certain it is that we are 
producing no text-books in which the theory is pre- 
sented in the delightful style which characterizes 
many of the French works (for example, that of 
Bourlet), or those of the recent Italian school (like 
Pincherle’s handbooks), or, indeed, those of the conti- 
nental writers in general. ‘In short, the logic of the 
subject, which, both educationally and scientifically 
speaking, is the most important part of it, is wholly 
neglected. The whole training consists in example 
grinding. What should have been merely the help 
to attain the end has become the end itself. The 
result is that algebra, as we teach it, is rules, whose ob- 
ject is the solution of examination problems. ... The 
result, so far as problems worked in examinations go, 
is, after all, very miserable, as the reiterated com- 


1 Henrici, O., Presidential address, British Association, Section A, 1883. 


220 THE TEACHING OF ELEMENTARY MATHEMATICS 


plaints of examiners show; the effect on the ex- 
aminee is a well-known enervation of mind, an 
almost incurable superficiality, which might be called 
Problematic Paralysis ——a disease which unfits a man 
to follow an argument extending beyond the length 
of a printed octavo page. ... Against the occa- 
sional working and propounding of problems as an 
aid to the comprehension of a subject, and to the 
starting of a new idea, no one objects, and it has 
always been noted as a praiseworthy feature of Eng- 
lish methods, but the abuse to which it has run is 
most pernicious.” } 

The interpretation of solutions — Algebra is generous, 
says D’Alembert; it often gives more than is asked.” 
And it is one of the mysteries which teachers and 
text-books usually draw about the science, that some 
of the solutions of the applied problems are not 
usable, are meaningless. 

But there should be no mystery about this. It is a 
fact, easily explained, that it is not at all difficult to put 
physical limitations on a problem that shall render the 
result mathematically correct but practically impossible. 
For example, if I can look out of the window 9 times in 
2 seconds, how many times can I look out in 1 second, 
at the same rate? The answer, 44 times, is all right 


1 Chrystal, Presidential address of 1885. 
2L’algébre est généreuse; elle donne souvent plus qu’on ne lui 
demande. 


TYPICAL PARTS OF ALGEBRA | 221 


mathematically, but physically I cannot look out half a 
time. Similarly, if 5 men are to ride in 2 carriages, 
how many will go in each, the carriages to contain the 
same number? Mathematically the solution is simple, 
but a physical condition has been imposed, “the car- 
riages to contain the same number,” which makes the 
problem practically impossible. A few such absurd 
cases take away all the mystery attaching to results of 
this nature, and show how easy it is to impose restric- 
tions that exclude some or all results. 

For example, the number of students in a certain 
class is such as to satisfy the equation 227-334 
—140=0; how many are there? The conditions of 
the problem are such as to make one root, 20, legiti- 
mate, but the other, —4, meaningless. Algebra 
has been generous; it has given more than was 
asked. 

Consider also the problem, A father is 53 years old 
and his son 28; after how many years will the father 
be twice as old as the son? From the equation 
53 +4+=2(28+-24) we have r=— 3. We are now 
under the necessity of either (1) interpreting the ap- 
parently meaningless answer, — 3 years after this time, 
or (2) changing the statement of the problem to avoid 
such a result. Either plan is feasible. We may in- 
terpret “— 3 years after” as equivalent to “3 years 
before,” which is entirely in accord with the notion 
of negative numbers; or we may change the problem 


222 THE TEACHING OF ELEMENTARY MATHEMATICS 


to read, “‘How many years ago was the father twice as 
old as the son.” Most algebras require this latter 
plan, one inherited from the days when the negative 
number was less understood than now. 

“Unlike other sciences, algebra has a special and 
characteristic method of handling impossibilities. If 
this problem of algebra is impossible, if that equation 
is insoluble, instead of hesitating and passing on to 
some other question, algebra seizes these solutions and 
enriches its province by them. The means which it 
employs is the symbol.”1 The symbol “— 3,” for the 
number of years after the present time, without sense 
in itself, is seized and turned into a means for enriching 
the domain of algebra by the introduction and interpre- 
tation of negative numbers. 

The further interpretation of negative results, and 
the discussion of the results of problems involving lit- 
eral equations, is a field of considerable interest and 
value; but since most text-books furnish a sufficient 
treatment of the subject, it need not be considered 
here. 

Conclusion —.The few topics mentioned in this chap- 
ter might easily be extended. It would be suggestive 
to dwell upon the absurdity of drilling a pupil upon the 
two artificially distinct chapters on surds and fractional 
exponents, as our ancestors used to separate the “rule 
of three”’ from proportion — matters explainable only 


1 De Campou. 


TYPICAL PARTS OF ALGEBRA 223 


by reviewing their history. The theory of fractions, 
the common fallacy in the proof of the binomial the- 
orem for general exponents, the use of determinants, 
the complete explanation of division or involution, the 
questions of zero, of infinity, and of limiting values — 
these and various other topics will suggest themselves 
as worthy a place in a chapter of this kind. But the 
limitations of this work are such as to exclude them. 
The topics already discussed are types, and it is hoped 
that they may lead some of our younger teachers of 
algebra to see how meagre is the view offered by many 
of our elementary text-books, how much interest can 
easily be aroused by a broader treatment of the simpler 
chapters, and how necessary it is to guard against the 
dangers of the slipshod methods and narrow views 
which are so often seen in the schools. As algebra 
is often taught, there is force in Lamartine’s accusa- 
tion, that mathematical teaching makes man a machine 
and degrades thought,! and there is point to the 
French epigram, “One mathematician more, one man 
less.” ? 


1 T’enseignement mathématique fait ’homme machine et dégrade la 
pensée. Rebiére’s Mathématiques et mathématiciens, p. 217. 

2 Un mathématicien de plus, un homme de moins. Dupanloup. Quoted 
in Rebiére, ib., p, 217. 


CHAPTER IX 
THE GROWTH OF GEOMETRY 


Its historical position — Roughly dividing elementary 
mathematics into the science of number, the science 
of form, and the science of functions, the subject has 
developed historically in this order. Hence the chrono- 
logical sequence would lead to the consideration of 
geometry before algebra, not only in the curriculum, 
but in a work of this nature. The somewhat closer 
relation of arithmetic and algebra, however, explains 
the order here followed, if explanation is necessary 
for a matter of so little moment. 

Reserving for the following chapter, as was done 
with algebra, the question of the definition of geom- 
etry, we may consider by what steps the science as- 
sumec its present form. We shall thus understand 
more clearly the limitations which the definition will be 
seen to place upon the subject, we shall see the trend 
which the science is taking, and we shall the more 
plainly comprehend the nature of the work to be 
undertaken by the next generation of teachers. 

The dawn of geometry— The world has always 
tended to deify the mysterious. The sun, the stars, 
fire, the sea, life, death, number —these have all 

224 


THE GROWTH OF GEOMETRY 225 
; 


played parts in the great religious drama. Whether it 
be that the plains of Babylon were especially adapted 
to the care of flocks, or that the purity of the Egyp- 
tian atmosphere led to the study of the heavenly 
bodies, or that both of these causes played their parts, 
certain it is that in Mesopotamia and along the-Nile 
a primitive astronomy developed at an early period and 
took its place as a part of the store of ancient reli- 
gious mysteries. With it went some rude knowledge of 
geometry, the demands of practical life also creating from 
time to time an empirical science of simple mensuration. 

Thus among the Babylonians we find the circle of 
the year early computed at 360 days (whence the circle 
was divided into that number of degrees), and later, 
as astronomical observation improved, at more nearly 
the correct number.) The Babylonian monuments so 
often picture chariot wheels as divided into sixths, that 
it is probable that the method of dividing the circum- 
ference into sixths by means of striking circles was 
early known, a method which carries with it the inscrip- 
tion of the regular hexagon. This would show that the 
circumference is a little more than 67 or 3d, but 7 
seems generally to have been taken as 3 by them and 
their neighbors.” 


1 Hankel, Zur Geschichte der Mathematik, p. 71, for the pre-scientific 
geometry. 

21 Kings vii, 23; 2 Chron. iv, 2. “What is three handbreadths 
around is one handbreadth through.” Talmud. 


Q 


226 THE TEACHING OF ELEMENTARY MATHEMATICS 


The Egyptians were particular as to the proper 
orientation of their temples, a custom still considered of 
moment by a large part of the religious world. The 
meridian line was established by the pole star, and for 
the east and west line the temple builders were early 
aware of a rule still used by surveyors in laying off a 
perpendicular. The present plan is to take eight links 
of a surveyor’s chain, place the ends of the chain four 
links apart, and stretch it with a pin at the fifth link, 
this forms a right-angled triangle with sides 3, 4,5. The 
Egyptians did the same in building their temples, and 
the harpedonapiae or “‘rope-stretchers”’ laid out the plans, 
as a civil engineer lays out those for a building to-day. 

The scholars of the Nile valley also possessed some 
knowledge of the rudiments of trigonometry,* and 
their approximation to the value of a was not im- 
proved for many centuries. Ahmes gave the value 
mw = (18)? = 3.1605, a remarkably good approximation 
for a period when geometry was little more than men- 
suration. He was not so fortunate in all of his rules, 
for example in the one for finding the area of an 
isosceles triangle, which required the multiplication 
of the measure of half the base by that of one of 
the equal sides. 


1 This interpretation of the Greek arpedonaptae is one of Professor 
Cantor’s ingenious discoveries. Cantor, I, p. 62. 

2A brief summary is given in Gow, History of Greek Mathematics, 
p. 128. 


THE GROWTH OF GEOMETRY 227 
t 


The indebtedness of the Greeks, who were the 
next to take up geometry, to the Egyptians is well 
summarized by Gow: “It remains only to cite the 
universal testimony of Greek writers, that Greek geom- 
etry was, in the first instance, derived from Egypt, 
and that the latter country remained for many years 
afterward the chief source of mathematical teaching. 
The statement of Herodotus on this subject has 
already been cited. So also in Plato’s ‘Pheedrus,’ 
Socrates is made to say that the Egyptian god Theuth 
first invented arithmetic and geometry and astronomy. 
Aristotle also (‘Metaphysics,’ I, 1) admits that geom- 
_ etry was originally invented in Egypt; and Eudemus 
expressly declares that Thales studied there. Much 
later Diodorus (70 B.c.) reports an Egyptian tradition 
that geometry and astronomy were the inventions of 
Egypt, and says that the Egyptian priests claimed 
Solon, Pythagoras, Plato, Democritus, Cenopides of 
Chios, and Eudoxus as their pupils. Strabo gives 
further details about the visits of Plato and Eudoxus. 

Beyond question, Egyptian geometry, such as 
it was, was eagerly studied by the early Greek phi- 
losophers, and was the germ from which in their hands 
grew that magnificent science to which every English- 
man is indebted for his first lessons in right seeing and 
thinking.” } 

The Greeks were, however, the first to create a 


1 History of Greek Mathematics, p. 131. 


228 THE TEACHING OF ELEMENTARY MATHEMATICS 


science of geometry. Thales (— 640, — 548), having 
through trade secured the financial means for study, 
travelled in Egypt for the purpose of acquiring the 
mathematical lore of the priests, giving quite as much 
as he received, and finally established a school in 
Asia Minor, where the first important scientific in- 
vestigations in geometry were made. 

The most noted pupil of Thales was Pythagoras, 
who was with him for a short time at least and who 
was advised by him to continue his studies in Egypt. 
The school which Pythagoras afterward opened in 
Croton, in Southern Italy, was one of the’ “most 
famous of all antiquity, and here geometry was seri- 
ously cultivated. Here were proved the following 
propositions, among others: the plane about a point 
is filled by six equilateral triangles, four squares or 
three regular hexagons; the sum of the interior 
angles of a triangle is two right angles; the sum of 
the squares on the sides of a right-angled triangle 
equals the square on the hypotenuse, a fact known to 
the Egyptians but first proved by the Pythagoreans. 

From now on until the third century before Christ 
Greek geometry passed through its golden age.} 


1 For detailed notes as to the discoveries of the Greeks see Allman, 
G. J. Greek Geometry from Thales to Euclid; Bretschneider, Die 
Geometrie und die Geometer vor Eukleides, Leipzig, 1870; Gow, J., 
History of Greek Mathematics, Cambridge, 1884; Beman and Smith’s 
translation of Fink’s History of Mathematics, Chicago, 1900; Chasles, 


THE GROWTH OF GEOMETRY 229 


The principal discoveries in elementary geometry 
were made in the two centuries from — 500 to — 300, 
and were crystallized in logical form by Euclid, who 
taught in the famous school at Alexandria about 
— 300. During this period, owing to the vast extent 
of the field opened up by the study of conic sections, 
Plato (— 429, — 348) placed a definite limit upon elemen- 
tary geometry, allowing only the compasses and the 
unmarked straight-edge as instruments for the con- 
struction of figures. 

So complete as a specimen of logic was Euclid’s 
treatment of elementary geometry, that it has been 
used as a text-book, with slight modifications, for 
over two thousand years. This use has not, however, 
been general. Indeed, it has needed the exertions of 
men like Hoiiel in France and Loria! in Italy, and 
other Continental writers, to recall from time to time 
the merits of Euclid to the educational world. But 
in England Euclid still holds a sway that is prac- 
tically absolute.? 

The influence of the Greek writers is still seen in the 
M., Apercu historique sur Vorigine . . . de Géométrie, Paris, 2. éd., 1875; 
and of course Cantor and Hankel. 

1 Della varia fortuna di Euclide in relazione con i problemi dell’ In- 
segnamento Geometrico Elementare, Rome, 1893. 

2'Teachers who care to enter into the merits of the controversy over 
Euclid may make a pleasant beginning, and at the same time may 
see the mean between Dodgson the mathematician and Carroll the 


writer of children’s stories (as Alice in Wonderland) by reading Dodg- 
son, C. L., Euclid and his Modern Rivals, London. 


230 THE TEACHING OF ELEMENTARY MATHEMATICS 


nomenclature of the science the world over. Because 
the ancients had no printing, and found it convenient 
to have the rolls, which made their volumes, somewhat 
brief, the word “book” came to apply to part of a 
treatise. Thus we have the books of the “ Atneid,” of 
the “Iliad,” and of treatises on geometry, astronomy, 
etc. The word has been preserved in the divisions of 
most elementary geometries as a matter of interesting 
history. Thus Euclid’s first book is chiefly upon 
straight lines and the congruence of rectilinear figures; 
the second is devoted to the next subject of which the 
student has already some knowledge — rectangles; the 
third to circles, and so on. With doubtful judgment 
some of our modern writers have followed Legendre in 
reversing the order in the second and third books, 
placing circles before rectangles, the less known and 
more difficult concept before the more familiar and 
simple. 

Many other words, unlike “book,” are distinctly 
Greek, as, for example, “theorem,” ‘‘axiom,” ‘“ scho- 
lium” (happily going out of fashion), “trapezoid,” 
“parallelogram,” “parallelepiped” (often given the 
unscientific spelling “parallelopiped’’), ‘ hypotenuse”’ 
(still occasionally spelled with an %, though unscien- 
tifically so), etc. In many cases, however, the Latin 
forms have displaced the Greek, as in “triangle” 
(rather more Latin than Greek), “ quadrilateral,” 
“base; circumierence,e.* verttex, | at: SUtl ace Mace 


THE GROWTH OF GEOMETRY 231 


After the death of Archimedes (—212), to whom we 
owe the first fruitful scientific attempts at the mensura- 
tion of the circle, geometry seems to have exhausted 
itself. Excepting a few sporadic discoveries, it remained 
stagnant for nearly two thousand years. It was not 
until the seventeenth century that any great advance 
was made, a century which saw the discovery of analytic 
geometry at the hands of Descartes, the revival of pure 
geometry through the labors of Pascal and his contem- 
poraries, and which saw but failed to recognize the 
foundation of projective geometry in the works of 
Desargues. . 

Recent geometry — The nineteenth century has seen 
a notable increase of interest in the geometry of the 
circle and straight-edge, a geometry which can, how- 
ever, hardly be called elementary in the ordinary sense. 
France has been the leader in this phase of the subject, 
with England and Germany following. Carrying out 
the suggestion made by Desargues in the seventeenth 
century, Chasles, about the middle of the nineteenth cen- 
tury, developed the theory of anharmonic ratio, making 
this the basis of what may be designated modern geom- 
etry. Brocard, Lemoine, and Neuberg have been largely 
instrumental in creating a geometry of the circle and 
the triangle, with special reference to certain interesting 
angles and points. How much of all this will find its 
way into the elementary text-books of the next genera- 
tion, replacing, as it might safely do, some of the work 


232 THE TEACHING OF ELEMENTARY MATHEMATICS 


which we now give, it is impossible to say. The teacher 
who wishes to become familiar with the elements of 
this modern advance could hardly do better than read 
Casey’s Sequel to Euclid. 

Along more advanced lines the progress of geometry 
has been very rapid. The labors of Mobius, Plicker, 
Steiner, and Von Staudt, in Germany, have led to regions 
undreamed of by the ancients. This work is not, how- 
ever, in the line of elementary geometry, and therefore 
has no place in the present discussion.? 

Among the improvements which affect the teaching 
of the elementary geometry of to-day, a few deserve 
brief mention. Among these is the contribution of 
““Mobius on the opposite senses of lines, angles, sur- 
faces, and solids; the principle of duality as given by 
Gergonne and Poncelet; the contributions of De Mor- 
gan to the logic of the subject; the theory of trans- 
versals as worked out by Monge, Brianchon, Servois, 
Carnot, Chasles, and others; the theory of the radical 
axis, a property discovered by the Arabs, but intro- 
duced as a definite concept by Gaultier (1813) and 
used by Steiner under the name of ‘line of equal 
power’; the researches of Gauss concerning inscrip- 
tible polygons, adding the 17- and 257-gon to the list 
below the 1000-gon; ... and the researches of Muir 
on stellar polygons.... 

1London, fifth edition, 1888. 


2¥For a brief review of the subject, see the author’s article in Merriman 
and Woodward’s Higher Mathematics, New York, 1896, p. 558. 


THE GROWTH OF GEOMETRY 233 


“In recent years the ancient problems of trisecting 
an angle, doubling the cube, and squaring the circle 
have all been settled by the proof of their insolubility 
through the use of compasses and straight-edge.” ! 

Non-Euclidean geometry —‘‘The non-Euclidean ge- 
ometry is a natural result of the futile attempts which 
had been made from the time of Proklos to the opening 
of the nineteenth century to prove the fifth postulate 
(also called the twelfth axiom, and sometimes the elev- 
enth or thirteenth) of Euclid.” This is essentially the 
postulate that through a point one and only one line can 
be drawn parallel to a given line. The first scientific 
investigation of this part of the foundation of geometry 
was made by Saccheri (1733). The matter was brought 
to its final stage by Lobachevsky and Bolyai about 1825, 
and the result is a perfectly consistent geometry denying 
the validity, or the necessity, of the postulate in ques- 
tion.” 

1Smith, D. E., History of Modern Mathematics, in Merriman and 
Woodward’s work cited, p. 564. Onthe impossibility of solving the prob- 
lems mentioned, see Beman and Smith’s translation of Klein’s Famous 


Problems of Elementary Geometry, Boston, 1896. 
2 Smith, D. E., History of Modern Mathematics, p. 565. 


CHARTERS 


WHAT IS GEOMETRY? GENERAL SUGGESTIONS FOR 
TEACHING 


Geometry defined — The etymology of a word is often 
far from giving its present meaning. We have already 
seen this in the case of ‘‘algebra” and “algorism” 
(p. 151). Geometry means earth-measure (yf, the earth, 
+ petpetv, to measure), and probably took this name be- 
cause, in its prescientific stage, it was what we would 
now call by the unexpressive term “surveying.” It 
came to mean, among the Greeks, the science of fig- 
ures or of extent, and this general idea still obtains. 

More specifically we may say: “By the observation 
of objects about us we arrive at the concept of the space 
in which we live and in which these objects have a cer- 
tain extent. Weare aware at the same time that they 
have a form. These forms are infinitely varied, but 
certain of them strike us by their regularity.’ This 
regularity is rather apparent than real, and the appear- 
ance leads us to make certain abstractions, as of straight 
line, circle, square, etc., forms not met in nature. “Just 
as the abstractions made concerning collections of 
objects? are the basis of our arithmetical ideas, so the 


1Laisant, p. 89. 2See p. 100. 


234 


WHAT IS GEOMETRY 235 


abstractions made concerning forms are the origin of 
our conceptions of geometry.”! Hence the science of 
geometry is the science of certain abstractions which the 
mind makes concerning form. As Laplace says: “In 
order to understand the properties of bodies, we have 
first to cast aside their particular properties and to see 
in them only an extended figure, movable and impene- 
trable. We must then ignore these last two general 
properties and consider the extent only asa figure. The 
numerous relations presented under this last point of 
view form the object of geometry.” ? 

Elementary geometry, however, limits itself to com- 
paratively few of these forms. As already stated, the 
great field opened by the study of conics and higher 
plane curves led Plato to limit elementary plane 
geometry to those figures which can be constructed by 
the use of the compasses and the unmarked straight- 
edge. As solid geometry has gradually developed, it 
has been looked upon as limited to figures. analogous 
to those of plane geometry, the sphere analogous to 
the circle, the plane to the straight line, etc., with 
the addition of the prism, pyramid, cone, and cylin- 
der. Euclid, caring little for the practical demands 
of mensuration, paid almost no attention to solid 
geometry; but the subject has assumed much prom- 
inence in the nineteenth century, without, however, 


1Laisant, p. 89. 
2Dauge, F., Méthodologie, p. 161. 


236 THE TEACHING OF ELEMENTARY MATHEMATICS 


having its limits clearly defined. For example, whether 
a cone with a non-circular directrix shall be admitted 
is an unsettled question; for purposes of simple men- 
suration of volume it might deserve a place, but hardly 
so unless the mensuration of a non-circular curvilinear 
plane figure (its base) is also admitted. 

Limits of plane geometry — But elementary geom- 
etry is not only quite uncertain with respect to the 
extent of the portion devoted to solids; the recent 
additions to plane geometry, referred to in Chapter 
IX, have made the limits of that portion of the 
science, as. to its “elements,” even more undethned. 
With the recent ‘geometry of the triangle,’ as it 
is sometimes called, the extent of the subject is far 
beyond the possibilities of the secondary curricu- 
lum. It cannot all be excluded, for we have long 
since introduced the notions of orthocentre, centroid, 
ex-centre, etc., but just what shall be admitted by 
the next generation is quite uncertain, as would be 
expected in view of the fact that the development 
is so recent. Suffice it to say that at present there is 
no general agreement as to what constitutes element- 
ary geometry, save this—that it shall cover substan- 
tially the ground of Euclid’s “ Elements,” plus a little 
work on loci, the mensuration of the circle, and the men- 
suration of certain common solids. From this state- 
ment, the futility of attempting a scientific definition of 
the elementary geometry of the schools is apparent. 


WHAT IS GEOMETRY 237 


The reasons for studying geometry, as for studying 
arithmetic, are twofold. We have the practical side 
of the subject in simple mensuration, and we have 
the culture side in the logic which enters into it to 
such a marked degree. 

The most practical part of mensuration is usually 
taught in connection with arithmetic, formerly by 
mere rule, now with the models in hand and with a 
semi-scientific deduction of a few necessary formulae. 
To drop the science there, would be to lose its chief 
value, to do what the English schools do with solid 
geometry —a mistake also often made in our Eastern 
states, though not in the West. The danger of doing 
nothing with solid geometry save in the way of men- 
suration, is suggested by Professor Henrici in these 
words (referring to the English schools): ‘Most of 
all, perhaps, solid geometry has suffered, because 
Euclid’s treatment of it is scanty, and it seems, 
almost incredible that a great part of it—the men- 
suration of areas of simple curved surfaces and of 
volumes of solids—is not included in ordinary school 
teaching. The subject is, possibly, mentioned in 
arithmetic, where, under the name of mensuration, a 
number of rules are given. But the justification of 
these rules is not supplied, except to the student 
who reaches the application of the integral calculus; 
and what is almost worse is that the general relation 
of points, lines, and planes, in space, is scarcely 


238 THE, TEACHING OF ELEMENTARY MATHEMATICS 


touched upon, instead of being fully impressed on 
the student’s mind.” ! 

The culture value, which is almost the only one 
which formal, demonstrative geometry has, includes two 
phases. In the first place, we need to know geometry 
for general information, because the rest of the world 
knows something of it. It must be admitted, however, 
that this is not a very determining reason, for it is one 
which would justify keeping any traditional subject in 
the curriculum. 

The second and vitally important culture phase is 
that of the logic of geometry. Before Euclid, probably 
most of his propositions were known; but it was he 
who arranged them in the order and with the demon- 
strations which have made his work one of the most 
admired specimens of logic that have ever been pro- 
duced. And this logic has given added significance 
and beauty to the truths themselves. ‘They enrich us 
by our mere contemplation of them. In this connection 
I wish to quote the beautiful poem ‘Archimedes and 
the Student,’ by Schiller : 


“To Archimedes once came a youth, who for knowledge was thirst- 
ing, 
Saying, ‘Initiate me into the science divine, 
Which for my country has borne forth fruit of such wonderful 
value, 
And which the walls of the town ’gainst the Sambuco protects.’ 


1 Presidential address, 1883. 


WHAT IS GEOMETRY 239 


‘Call’st thou the science divine ? It is so,’ the wise man responded ; 

‘But it was so, my son, ere it availed for the town. 

Would’st thou have fruit from her, only? even mortals with that 
provide thee ; 

Would’st thou the goddess obtain? seek not the woman in her !’”! 


Here, then, is the dominating value of geometry, its 
value as an exercise in logic, as a means of mental 
training, “as a discipline in the habits of neatness, 
order, diligence, and, above all, of honesty. The 
fact that a piece of mathematical work must be definitely 
right or wrong, and that if it is wrong the mistake can 
be discovered, may be made a very effective means of 
conveying a moral lesson.” Without this aim well 
fixed in mind, the teacher is like a i.ariner without a 
compass; he knows not whither he is to go; or, if he 
has some confused idea of the haven, he has not the 
means to find his way thither. 

Having now considered the nature of elementary 
geometry, and the reasons for teaching the science, 
the question arises as to the general method of pre- 
senting it. 

Geometry in the lower grades — While educators differ 
materially as to the method of presenting the subject of 
demonstrative geometry, this being still an open question 
for the coming generation to consider, it is generally 


1 Schwatt, I. J., Some Considerations showing the Importance of Mathe- 
matical Study, Philadelphia, 1895. 
2 Mathews, G. B., in The School World, Vol. I, p. 129 (April, 1899). 


240 THE TEACHING OF ELEMENTARY MATHEMATICS 


agreed that some of the elementary concepts of the 
science should be acquired in the lower grades. This 
view was long ago held by Rousseau. ‘TI have said,” 
he remarks, “that geometry is not adapted to children ; 
but this is our fault. We seem not to comprehend that 
their method is not ours, and that what should be for us 
the art of reasoning should be for them merely the art 
of seeing. Instead of thrusting our method upon them, 
we would do better to adopt theirs. ... For my pupils, 
geometry is merely the art of handling the rule and 
compasses.’ ! Lacroix, one of the best teachers of 
mathematics at the opening of the nineteenth century, 
recognized the same principle when he said: “‘ Geometry 
is possibly of all the branches of mathematics that which 
should be understood first. It seems to me a subject 
well adapted to interest children, provided it is presented 
to them chiefly with respect to its applications. ... The 
operations of drawing and of measuring cannot fail to 
be pleasant, leading them, as by the hand, to the science 
of reasoning.” Such was also the scheme laid out by 
the mathematician Clairaut and approved by Voltaire, 
but in practice it has not been systematically followed 
by the teaching profession. | 
Laisant, whose rank as a mathematician and an 


1 Rébiére, A., Mathématiques et mathématiciens, p. 103. 

2 His Essais sur l’enseignement en général, et sur celui des mathé- 
matiques en particulier, Paris, 1805, was one of the earliest works of any 
value on the teaching of mathematics. 


WHAT IS GEOMETRY 241 


educator justifies the frequent reference to his name, 
thus expresses his views: “The first notions of ge- 
ometry should be given to the child along with the 
first notions of algebra, following closely upon the 
beginning of theoretical arithmetic (larithmétique 
raisonnée). But just as there must be a preliminary | 
preparation for arithmetic, namely practical calcula- 
tion, so theoretical geometry should be preceded by 
the practice of drawing. The habit acquired in 
childhood, of drawing figures neatly and with sen- 
sible exactness, would be of great assistance later in 
the development of the various chapters of geometry. 
The one who defined geometry as the art of correct 
reasoning on bad figures, was altogether wrong. We 
‘never reason save on abstractions, and figures are 
never exact; but when the inaccuracy is too manifest, 
when the drawings are poorly executed and appear 
confused, this confusion of form readily leads to that 
of reasoning and tends to conceal the truth. Indeed | 
there are cases where a poorly drawn figure leads 
by logical reasoning to manifest absurdities! The 
first education in geometry should therefore be under- 
taken, as in the case of practical computation, with 
the child who knows how to read and write the 
language —that is, who knows drawing. ... Advan- 
tage should be taken in this drawing of figures, to 


1 Two interesting illustrations of this fact are given in Ball’s Mathe- 
matical Recreations, London, 1892, p. 32. 


R 


242 THE TEACHING OF ELEMENTARY MATHEMATICS 


give to the child the nomenclature of a large number 
of geometric concepts, but always without any formal 
definitions.” } 

The views of Hoiiel, one of the best teachers of 
the last generation in France, also deserve recogni- 
tion, “Let us. imagine,” he says, “the possibility 
of a graduated teaching of elementary geometry 
carried on at every step according to one invariable 
plan, always governed by the rules of severe logic, 
but with the difficulties always commensurate with 
the development of the pupil’s mind. For such a 
scheme the study of geometry would need to be con- 
sidered from various points of view corresponding to 
the various degrees of initiation of the pupil. For 
beginners it would be necessary first of all to famil- 
larize them with the various geometric figures and 
their names, to lead them to know facts and to 
understand their more simple and immediate appli- 
cations to matters of daily life. We ought at first 
to multiply the axioms and to employ, in place 
of demonstrations, experimental truths, analogy, in- 
duction, always remembering that this method of 
treatment is essentially provisional. . .. The first 
teaching should be purely experimental, and little by 
little the pupil should come to see that all truths 
need not necessarily be derived from experience, but 
that some are consequences of a certain number of 


1 La Mathématique, p. 220. 


WHAT IS GEOMETRY 243 


others, a number which becomes smaller and smaller 
as one advances in the science until he reaches the 
fundamental axioms.” } 

The ideas above set forth are not the thoughts of 
mere theorizers; they have been carried out with 
more or less success in many European and Ameri- 
can schools. The outline of some of this work is 
given in the subsequent pages. It may, however, be 
said for the lower grades, in passing, that teachers 
should insist that none of the new schemes of draw- 
ing which apply for admission to the schools be 
lacking in this particular. The study of the common 
geometric forms in the early years is too valuable to 
be neglected. 

Intermediate grades— The next step in the work 
is taken in the so-called ‘‘grammar grades.” The 
mensuration of the common surfaces and solids should, 
of course, never be a matter of arbitrary rule. Our 
best text-books in elementary arithmetic at present 
give satisfactory development of the rules for all 
necessary cases not involving irrational numbers. A 
pair of shears and some cardboard enable the teacher 
to pass from the rectangle to the parallelogram, and 
thence to the trapezoid and the triangle, developing 
the formulae or rules with little difficulty. Similarly 
the formulae for the circle can be developed by cut- 
ting this figure into sectors which are approximately 


1 Rebiére, Mathématiques et mathématiciens, p. 102, 


244. THE TEACHING OF ELEMENTARY MATHEMATICS 


triangles. Only a little labor is needed to prepare 
pasteboard models of the most common geometric 
solids, and these, together with a pail of dry sand 
for filling some of them in comparing volumes, fur- 
nish the materials for developing the formulae for 
“measuring such bodies.! 

Nor should we regard this method of investigation 
unscientific. It merely follows the line of historic 
development, the line in which truth is first acquired 
by induction. Comte cites an interesting illustration 
of this method, showing the way in which Galileo 
determined the ratio of the area of an ordinary 
cycloid to that of the generating circle. ‘The geome- 
try of his time was as yet insufficient for the rational 
solution of such problems. Galileo conceived the 
idea of discovering that ratio by a direct experiment. 
Having weighed as exactly as possible two plates of 
the same material and of equal thickness, one of 
them having the form of a circle and the other that 
of the generated cycloid, he found the weight of the 
latter always triple that of the former ; whence. he 
inferred that the area of the cycloid is triple that of 
the generating circle, a result agreeing with the 
veritable solution subsequently obtained by Pascal and 


1 For directions as to this work see Beman and Smith’s Higher Arith- 
metic, Boston, 1896, p. 66. Reference should also be made to a valuable 
pamphlet by Professor Hanus, Geometry in the Grammar School, Boston, 


1893. 


WHAT IS GEOMETRY 245 


Wallis.” 1 It would be well, indeed, if we had even 
more of this induction along with our later demonstra- 
tive geometry. One of the common sources of failure, 
especially in the discovery of loci and the solution of 
certain other problems, is the inability of the pupil to 
make correct inductions, from carefully drawn figures, 

Along with this work in mensuration should con- 
tinue the geometric drawing begun in the earlier 
grades. The subject has been worked out with con- 
siderable success by several writers.? 

Spencer’s Inventional Geometry, while not an ideal 
text-book, was a noteworthy step in this direction of 
scientific induction based upon accurately drawn figures. 
Dr. Shaw, speaking of his experiments with children 
along the lines suggested by Spencer, says: “A few 
months’ work proved the incalculable value of inven- 
tional geometry in a school course of study; and eleven 
years’ experience in many classes and in different 
schools confirms that judgment. 

“In these classes the pleasure experienced in the 
study has made the work delightful both to teacher and 
to taught; and there has always been a continuous 


1 Philosophy of Mathematics, English by Gillespie, New York, 1851, 
p. 186. 

2 Spencer, W. G., Inventional Geometry, New York, 1876; Harms, 
Erste Stufe des mathematischen Unterrichts, II. Abt. 3. Aufl., Oldenburg, 
1878, along the same lines as a work by Gille (1854); Schuster, M., Auf- 
gaben fiir den Anfangsunterricht in der Geometrie, Program, Oldenburg, 
1897. Campbell, Observational Geometry, New York, 1899. 


246 THE TEACHING OF ELEMENTARY MATHEMATICS 


interest from the beginning to the end of the term. 
This pleasure and interest came, not from any environ- 
ment, not from the peculiar individuality of the class, 
but because the problems are so graded and stated that 
the pupil’s progress becomes one of self-development — 
a realization of the highest law in education. .. . 

“The pupil should not be told or shown, but thrown 
back upon himself; for, in inventional geometry, the 
knowledge is to be gained by growth and experience, 
through the powers he possesses and the method of 
acquirement peculiar to his mind. Occasionally the 
pupil is not a little baffled, and the skill of the teacher 
is put to its best test to gain the solution without show- 
ing or telling him. Telling or showing is the method of 
the instructor — not the teacher... . 

“ Inventional geometry should precede the demonstra- 
tive, so as to give the pupil many concepts to draw 
upon when he takes up syllogistic demonstration. De- 
monstrative geometry then becomes an easier subject, 
and he is surer of what he is doing, because he has 
more general notions as a basis.” 

Speaking of Spencer’s work, Mr. Langley, one of the 
best teachers of elementary mathematics in England, 
confirms the views already expressed: “ It has not been 
usual for students, at any rate in schools, to approach 
the study of geometry in this experimental way, though 
there have probably always been individual teachers 
who have used it to varying extents. Of late years, 


WHAT IS GEOMETRY 247 


however, —in fact since more attention has been given 
to the theory and practice of education, —it has been 
strongly advocated. My own experience confirms me 
day by day in the opinion that it is the best method 
for the majority of students, though a few may be 
able to dispense with it. 

“It has two advantages: (1) It leads to clear con- 
ceptions of the truths to be established; (2) it may be 
used to imtroduce the student naturally to a different 
method of establishing such truths—the deductive 
method.” ! 

In America Professor Hanus has been prominent 
in putting the work on a practical basis.2 He rec- 
ommends two recitation periods per week for the 
seventh and eighth grades, and one for the ninth, 
the periods to be at least thirty minutes long. The 
following are his guiding principles for teachers: 

“1, FE rly instruction in geometry should be ob- 
ject teaching. 

“2, The pupil should make and keep an accurate 
record of his observations, and of the definitions or 


1 Langley, E. M., How to learn Geometry, The Educational Review 
(London), Vol. VIII, O.S., p. 3. The subject is also discussed, with a brief 
list of German text-books, in Dressler’s Der mathematisch-naturwissen- 
schaftliche Unterricht an deutschen (Volksschullehrer-) Seminaren, in 
Hoffmann’s Zeitschrift, XXIII. Jahrg., p. 18. 

2 Outline of work in Geometry for the Seventh, Eighth, and Ninth 
Grades of the Cambridge Public Schools, Boston, 1893; Geometry in the 


Grammar School, Boston, 1893. 


248 THE TEACHING OF ELEMENTARY MATHEMATICS 


propositions which his examination of the object or 
objects has developed. 

“3. In all his work the pupil should be taught 
to express himself by drawing, by construction, and 
in words, as fully and accurately as possible. The 
language finally accepted by the teacher should be 
the language of the science, and not a temporary 
phraseology to be set aside later. 

“4. The pupil is to convince himself of geomet- 
rical truths primarily through measurement, drawing, 
construction, and superposition, not by a logical dem- 
onstration. But gradually (especially during the last 
year of the work) the pupil should be led to attempt 
the general demonstration of all the simpler propo- 
sitions. 

‘5. The subject should be developed by the com- 
bined: edfort of. teacher and) pupil, 7c. thes teacher 
and the pupil are to codperate in reconstructing the 
subject for themselves. This is best accomplished by 
skilful questioning wethout the use of a text-book con- 
taining the definitions, solutions, and demonstrations... . 

“6. The subject-matter of each lesson should be 
considered in its relation to life, ze. the actual 
occurrence in nature and in the structures of ma- 
chines made by man of the geometrical forms studied, 
and the application of the propositions to the ordi- 
nary affairs of life should be the basis and the 
outcome of every exercise... . 


WHAT IS GEOMETRY 249 


‘7, Accuracy and neatness are absolutely essen- 
tial in all work done by the pupils.” ! 

In Germany a course extending through what 
corresponds to our “grammar school” has been out- 
lined by several writers. Without going into details, 
the following course suggested by Rein may serve 
to show what ground the modern Herbartians pro- 
pose to cover. 

A. Geometric form (Geometrische Formenlehre). 

Fourth school year —The cube, square prism, ob- 
long prism, triangular prism, quadrangular pyramid. 
In addition to these solids the pupil considers the 
point, straight line, surface, direction, measurement of 
the straight line, the right angle and its parts, the 
square and its construction, the rectangle and its 
construction, the triangle, and the diagonals of the 
rectangle. 

Fifth school year —The hexagonal prism, octagonal 
prism, hexagonal and octagonal pyramid, truncated 
pyramid, cylinder, cone, truncated cone, and sphere. 
The following plane figures are also studied: the 
regular hexagon and octagon, the obtuse angle, the 
trapezoid and circle. 

B. Geometry. 


Sixth school year — Properties of magnitudes (Eigen- 





schaften, Gesetze, der Raumgrossen), constructions, and 
mensuration. Size and measurement of angles, the 


1 Hanus. The course is outlined in both pamphlets. 


250 THE TEACHING OF ELEMENTARY MATHEMATICS 


protractor. Division of angles. Kinds and properties 
of triangles and parallelograms, with constructions. 
Mensuration of surfaces, the square, rectangle, paral- 
lelogram, and triangle. The trapezoid. The circle, 
its sectors and segments, and the value of 7. Reg- 
ular polygons. 

Seventh school year — Measurement and drawing of 
solids. 

C. Practical geometry. 

Eighth school year —1. The congruence proposi- 
tions. 2. Similarity. 3. Pythagorean theorem. Appli- 
cations to practical mensuration. ! 

Demonstrative geometry — The next step brings the 
student to demonstrative geometry, the geometry of 
Euclid, or its equivalent. Here the educator is at 
once confronted by the question, When shall this 
work be begun? 

In England Euclid is begun at an age which sur- 
prises American educators. In the lycées of France 
and the Gymnasien (or Realschulen, etc.) of Germany, 
as well as in most of the other preparatory schools 
of Europe, demonstrative geometry, although not 
Euclid, also finds much earlier place than in America. 
With us the subject usually begins in the tenth or 
eleventh school year, and the ‘Committee of Ten”’ 
recommends no change in this plan. To begin a 


1 Rein, Pickel and Scheller, Theorie und Praxis des Volksschulunter- 
richts; Das vierte Schuljahr, 3. Aufl, Leipzig, 1892, p. 232. 


WHAT IS GEOMETRY 251 


work of the difficulty of Euclid any earlier than this 
will hardly be sanctioned by American teachers; the 
hard Euclidean method must change, or the subject 
must remain thus late in the curriculum. If the 
object were, as seems to be the case in England, 
to cram the memory for an examination, it could be 
attained here as easily as there. But the considerable 
personal experience of the writer, as well as the far 
more extended researches of others, convinces him 
that as a valuable training in logic, as a stimulus to 
mathematical study, and as a foundation for future 
research, the study of Euclid as undertaken in Eng- 
land is not a success.1 If one has any doubt as to 
this judgment, it should be removed by this recent 
testimony of Professor Minchin, a man _ thoroughly 
familiar with the system, and an excellent math- 
ematician and teacher in spite of the fact that he 
was brought up on Euclid. 

“Why, then,” he says, “is it that the teacher, when 
he comes to the teaching of Euclid, is confronted 
with such great difficulties that his belief in the 
rationality of human beings almost disappears with 
the last vestiges of that good temper which he him- 
self once possessed? The reason is simply that 


1 Holzmiiller, G., Notwendigkeit eines propddeutisch-mathematischen 
Unterrichts in den Unterklassen héherer Lehranstalten vor dem wis- 
senschaftlich-systematischen, Hoffmann’s Zeitschrift, XX VI. Jahrg., p. 321, 


334: 


252 THE TEACHING OF ELEMENTARY MATHEMATICS 


Euclid’s book is not suitable to the understanding of 
young boys. It fails signally as regards both its lan- 
guage and its arrangement. ... For myself, I con- 
fess that, to the best of my belief, I had been through 
the six books of Euclid without really understanding 
the meaning of an angle.” 

If, however, a series of text-books should appear 
which carried the essential part of the first three 
books of Euclid along with the arithmetic and alge- 
bra work of the seventh, eighth, and ninth school 
years, thus connecting the severe demonstrative ge- 
ometry with that outlined for the lower grades, it 
would then be entirely feasible to begin demonstra- 
tive geometry earlier than now. We have, however, 
no such books in English, at least none which have 
succeeded in any such way as Holzmiiller’s excel- 
lent series has in Germany. That a child in the 
seventh grade can handle the pons asinorum of 
Euclid quite as easily as the problems he often 
meets in arithmetic, has been shown too often to 
admit of dispute. But in America we have been 
showing this only in sporadic cases, without formu- 
lating a well-ordered scheme of work which should. 
spread the geometry out, along with the algebra and 
the arithmetic. It is reasonable to expect that this 


1The School World (London), Vol. I, 1899, p. 161. 
2 In this connection the conclusion of Holzmiiller’s article mentioned 
on p. 251 is of interest. 


WHAT IS GEOMETRY 253 


plan will materialize before many years, through the 
skilful labors of some educated writer of a series 
of text-books. “That algebra, arithmetic, and geome- 
try should be taught side by side is not merely use- 
ful; it is indispensable for maintaining that unity 
and coordination in mathematics, without which the 
science loses all interest and value. A boy who has 
taken his arithmetic first, and then his algebra, and 
then his geometry, has his mental powers less de- 
veloped (/’esprit moins formé) than they would have 
been with three or four years of parallel teaching 
intelligently pursued.” ? 

Naturally a child loves form quite as much as 
number. Practically he needs number more often, 
and hence the elements of computation have been 
taught to him at an early age. But when we come 
into the theoretical part of arithmetic — greatest com- 
mon divisor, roots, proportion, etc.—it is merely an 
accident (historically explainable) that education has 
carried the child to the study of number and func- 
tions rather than to the study of form. 

Hence in general it may be said that the study of 
demonstrative geometry might profitably begin earlier 
than it does in the American schools, but that this 
would require, for the best results, a style of presen- 
tation quite different from that of Euclid or his 
modern followers. 


1 Laisant, La Mathématique, p. 227. 


254 THE TEACHING OF ELEMENTARY MATHEMATICS 


The use of text-books— But taking the curriculum 
as it stands in America at present, what general 
method of presentation shall be followed, and what 
kind of text-book shall be recommended? The great - 
majority of teachers take some text-book, require the 
pupils to prove the theorems substantially as therein 
set forth, and demand the demonstration of a con- 
siderable number of propositions which the English 
call “riders’»—aname quite as good (and bad) as 
our “original exercises.” The result is a tendency to 
fall into the habit of merely memorizing the solutions, 
thus losing sight of the greatest value of the subject 
—the training which it gives in logic. 

To avoid this danger, numerous plans have been 
devised. One is that of dictating the propositions, 
giving a few suggestions, and requiring the pupil to 
work out his own proofs. This plan, however, is 
open to several objections so serious as to condemn 
it in the minds of most educators. In the first place 
there is a great waste of time in the dictation of 
notes —a return to medizvalism. Furthermore, if the 
usual sequence of propositions is varied, few teachers 
have the ability to make this variation without destroy- 
ing something of the logic or symmetry of the sub- 
ject; if the usual sequence is followed, the pupil simply 
secures some text-book on geometry, often a poor one, 
and memorizes from that. Again, the pupil loses the 
advantage of having constantly before him a standard 


WHAT IS GEOMETRY 255 


of excellence in logic, in drawing, and in arrangement 
of work, and he fails to acquire the power to read 
and assimilate mathematical literature, a serious bar 
to his subsequent progress in more advanced lines. 

To meet the first of the above objections, the waste 
of time in dictation, text-books have been prepared 
containing merely the definitions, postulates, axioms, 
enunciations, etc. But while free from the first objec- 
tion, they are open to the others, and hence have met 
with only slight favor. 

Text-books have also been prepared which, in place 
of the proofs, submit series of questions, the answers 
to which lead to the demonstrations. This is the 
heuristic method put into book form; it substitutes a 
dead printed page for a live intelligent teacher. The 
substitution is necessarily a poor one, for the printed 
questions usually admit of but a single answer each, 
and hence they merely disguise the usual formal proof. 
They give the proof, but they give no model of a logical 
statement. 

The kind of text-book which the world has found 
most usable, and probably rightly so, is that which 
possesses these elements: (1) A sequence of proposi- 
tions which is not only logical, but psychological; not 
merely one which will work theoretically, but one in 
which the arrangement is adapted to the mind of the 
pupil; (2) Exactness of statement, avoiding such slip- 
shod expressions as, “A circle is a polygon of an in- 


256 THE TEACHING OF ELEMENTARY MATHEMATICS 


finite number of sides,” “ Similar figures are those with 
proportional sides and equal angles,” without other 
explanation; (3) Proofs given in a form which shal] 
be a model of excellence for the pupil to pattern after ; 
(4) Abundant exercises from the beginning, with prac: 
tical suggestions as to methods of attacking them; 
(5) Propzedeutic work in the form of questions or exer- 
cises, inserted long enough before the propositions 
concerned to demand thought—that is, not immedi- 
ately preceding the author’s proof. 

Such a book gives the best opportunity for success- 
ful work at the hands of a good instructor. But no 
book can ever take the place of an enthusiastic, re- 
sourceful teacher. In the hands of a dull, mechanical, 
gradgrind person with a teacher’s license, no book 
can be successful. The teacher who does not antici- 
pate difficulties which would otherwise be discouraging 
to the pupil, tempering these difficulties (but not wholly 
removing them) by skilful questions, is not doing the 
best work. On the other hand, the teacher who over- 
develops, who seeks to eliminate all difficulties, who 
does all of the thinking for the class, is equally at 
fault. Youth takes little interest in that which offers 
no opportunity for struggle, whether it be on the play- 
ground, in the home games of an evening, or in the 
classroom. 


CHAPTER :X!I 
THE BASES OF GEOMETRY 


The bases— Geometry as a science starts from cer- 
tain definitions, axioms, and postulates. It is hardly 
the province of this work to enter into a_ philosophi- 
cal discussion of the foundations upon which the 
science rests, first because such a discussion would 
require a volume of some size,! and also because 
practically the teacher is unable materially to change 
the definitions, axioms, and postulates of the text-, 
book which happens to be in the hands of his 
pupils. A brief consideration of these bases of the 
science may, however, be of service. 

The definitions of geometry occupy a position some- 
what different from that held by the definitions of 
algebra and arithmetic. There is not the same 
necessity for exactness in the definition of monomzal — 

1 The teacher may consult Dixon, E. T., The Foundations of Geometry, 
Cambridge, 1891 ; Russell, An Essay on the Foundations of Geometry, 
\ Cambridge, 1897; Poincaré, On the Foundations of Geometry, The 
' Monist, October, 1898; Hilbert, D., Grundlagen der Geometrie, in Fest- 
schrift zur Feier der Enthiillung des Gauss-Weber-Denkmals in Gottingen, 
Leipzig, 1899; Veronese, G., Fondamenti di Geometria, Padova, 1891; 


Koenigsberger, L., Fundamental Principles of Mathematics, Smithsonian 
Report, 1896, p. 93. 


s 257 


258 THE TEACHING OF ELEMENTARY MATHEMATICS 


as in that of right angle, for the latter is a control- 
ling factor in several logical demonstrations, while the 
former is not. In the same way more care must be 
shown in the definition of szmzlar figures than in that 
of semultaneous equations, of tsosceles triangle than of 
tncomplete quadratic, of parallelepiped than of dbinomzal ; 
not that all of these terms must not be well under- 
stood and properly used, and not that algebra is less 
exact than geometry, but that the geometric terms 
enter into logical proofs in such way as to make their 
exact statement a matter of greater moment. 

Hence the suggestions, already made in Chapter VIII 
upon accuracy of definition in algebra, apply with even 
greater force to geometry. Nor should the teacher 
attend so much to the idea that all the truth cannot 
be taught at once, as to acquire the dangerous habit 
of teaching partial truths ov/y, or (as too often happens) 
of teaching mere words, sometimes unintelligible, some- 
times wholly false. A few selections from our elemen- 
tary text-books will illustrate these points. 

We often see, for example, as a definition, “A 
straight line is the shortest distance between two 
points.” Now in the first place this is absurd, be- 
cause a lime is not adtstance,; distance is measured 
on a line, and usually on a curved one. Further- 
more, the statement merely gives one property of a 
straight line; it is a theorem, and by no means an 
easy one to prove. A definition should be stated in 


THE BASES OF GEOMETRY 259 


terms more simple than the term defined; but azstance 
is one of the most difficult of the elementary con- 
cepts to define Mathematicians have long since 
abandoned the statement. “It is a definition almost 
universally discarded, and it represents one of the 
most remarkable examples of the ‘persistence with 
which an absurdity can perpetuate itself through the 
centuries. In the first place, the idea expressed is 
incomprehensible to beginners, since it presupposes 
the idea of the length of a curve; and further, it is a 
case of reasoning in a circle (c’est un cercle vicieux), 
for the length of a curve can only be understood 
as the limit of a sum of rectilinear lengths. And 
finally, it is not a definition at all, but rather a 
demonstrable proposition.” ? 

The fact is, the concept stvazght line is element- 
ary; it is not capable of satisfactory definition, and 
hence it should be given merely some brief explana- 
tion. From Plato’s time to our own, attempts have 
been made to define such fundamental concepts as 
straight line and angle, but with no success. As 


1 Pascal’s rules for definitions are worthy of consideration in this 
connection : “(1) Do not attempt to define any terms so well known in 
themselves that you have no clearer terms by which to explain them ; 
(2) Admit no terms which are obscure or doubtful, without definition ; 
(3) Employ in definitions only terms which are perfectly well known 
or which have already been explained.” Rebiére, Mathématiques et 
mathématiciens, p. 23. 

2 Laisant, p. 223. 


260 THE TEACHING OF ELEMENTARY MATHEMATICS 


St. Augustine said of time, “If you ask me what 
it is, I cannot tell you; but if you do not ask me, 
I know too well.” And Pascal said of geometry: “It 
may be thought strange that geometry is unable to 
define any of its principal concepts; for it cannot 
define movement, or number, or space, and yet these 
are the very things which it considers most. It is 
not surprising, however, when we consider that this 
admirable science attaches itself only to the most 
simple concepts, and that the very quality which 
makes these worthy of being its objects renders them 
incapable of definition. Hence the inability to de- 
fine is rather a merit than a defect, since it arises 
not from the obscurity of the concepts, but rather 
from their extreme evidence.’’! 

Text-books are also liable to err on the side of 
redundancy in definition, as in the statement, “A 
rectangle is a parallelogram all of whose angles are 
right angles.” It would be thought absurd to say, 
“A rectangle is a four-sided parallelogram whose op- 


1 Rebiére, Mathématiques et mathématiciens, p. 16. For those who 
wish thoroughly to investigate the matter of the elementary definitions 
(straight line, angle, etc.), it will be of value to know that Schotten has 
compiled all of the typical definitions of these concepts which have ap- 
peared from the time of the Greeks to the present, and has set them forth 
with critical notes in his valuable treatise, Inhalt und Methode des plani- 
metrischen Unterrichts, Bd. I, 1890; Bd. II, 1893; Bd. III, in press. 
Professor Newcomb has also considered the matter briefly in the Appendix 
to his Geometry. 


THE BASES OF GEOMETRY 261 


posite sides are equal and parallel, and all of whose 
angles are right angles,’ because of the manifest 
redundancy. But if the definition is given at the 
proper place, it suffices to say, “If one angle of a 
parallelogram is a right angle, the parallelogram is 
called a rectangle.’ The same criticism applies to 
one of the common definitions of a square, “A rec- 
tangle whose sides are all equal”; it suffices if two 
adjacent sides are equal. The definition commonly 
given of similar figures is an illustration of the teach- 
ing of a half truth, the whole truth being thereby 
permanently excluded, and all this with no excuse. 
If a student beginning geometry were asked to name 
two similar figures, he would probably suggest two 
circles, or two spheres, or two straight lines, or two 
squares, and he would be right. But when he comes 
to the definition he finds that, of the four classes of 
figures named, only the squares are similar. It is, 
however, an easy matter to define similar systems of 
points, and then to say, “Two figures are said to 
be similar when their systems of points are similar,” 
thus admitting circles, spheres, similar cones, etc., all 
of which are excluded by the usual text-book defini- 
tion, and all of which deserve to be considered.! 

The introduction of the modern chapter on maxima 
and minima, in many of our elementary works, makes 


1 For further discussion see Beman and Smith’s New Plane and Solid 
Geometry, Boston, 1899, p. 182. 


262 THE TEACHING OF ELEMENTARY MATHEMATICS 


it worth while to say that the definition of maximum 
as the greatest value a variable can take, not only 
is misleading at the time, but also is conducive to sub- 
Sequent misunderstanding. Every teacher of geom- 
etry must be aware that, in general, a variable may 
have several maxima. 

The laxness of definitions which creeps into ele- 
mentary work is well illustrated in the case of the 
polyhedral angle. We not unfrequently find angle 
defined as “the difference of direction between two 
lines which meet” (a poor definition because the word 
angle is quite as elementary as the word direction), 
and the polyhedral angle defined as “the angle 
formed by three or more planes meeting in a point.” 
The absurdity appears when we substitute the defi- 
nition of angle for the word: “A polyhedral angle — 
is ‘the difference of direction between two lines which 


’ 


meet’ formed by three or more planes,” etc., and yet 
we teach mathematics as an exact science! This illus- 
tration is not a “man of straw’; one need not look 
far to find it. 

Axioms and postulates —In considering briefly the 
nature and the réle of the axioms and postulates of 
geometry, we may well begin by asking the meaning 
of the terms themselves. 

Of course it is true that these words mean to any 
generation just what the world at that time agrees 
they shall mean, and hence it is not a valid argu- 


THE BASES OF GEOMETRY 263 


ment to say that Euclid did not employ them in 
the sense understood by his early English trans- 
jlators. At the same time there has been, for a 
number of years, a feeling that the common defini- 
tions of postulate and axiom are absurd in statement 
and unscientific in thought, as well as unjustifiable 
historically. Heiberg,’ the most scholarly editor of 
the “/ements, has considered the matter very thor- 
oughly, and is convinced that Euclid used arzom for 
a general mathematical truth accepted without proof, 
and postulate for a similar truth of a geometric 
nature. Thus the statement, “If equals are added 
to equals the sums are equal,” is an axiom; but, 
“Through a given point but one line can be drawn 
parallel to a given line,’ is a postulate (not, how- 
ever, in Euclid’s language). The notion that an 
axiom is a “self-evident theorem,” and a postulate 
a problem too simple for solution, is therefore his- 
torically incorrect, as well as absurd in substance. A 
return to Euclid’s use of the words would seem desir- 
able, although the single word axzom for both classes 
would simplify matters. 

The definition of axzom as “a self-evident truth” has 
already been characterized as absurd. For what is self- 
evident to one mind is not at all so to another. It may 
be “self-evident” to a very good student that I is the 
only number whose cube is 1, until he tries cubing 


1 Euclidis elementa, Leipzig, 1883-88. 


264 THE TEACHING OF ELEMENTARY MATHEMATICS 


—i+1V—3; or that 2 is the only fourth root of 16, 
until some one suggests three others; or that ad must 
always equal da, until he studies quaternions or the 
theory of groups. The fact is, in geometry the word 
“axiom” is used merely to designate certain general 
statements the truth of which is assumed. Our senses 
seem to indicate that they are true; but whether true or 
false, we take them for granted and see whither they 
lead us. | 

Similarly, in geometry, with the word “postulate.” A 
postulate is a statement, referring to geometry, the truth 
of which is assumed. The statement may be true or it 
may be false, although our senses seem to indicate the 
former. That space is homogeneous seems true, but it 
may not be; but we assume it true and see whither we 
are led. So we may be able to draw, through a given 
point, more than one line parallel toa given line, although 
our senses, especially as biassed by our early training, 
seem to indicate not. But any one is entirely at liberty — 
to deny this or any other postulate, and to build up a 
logical geometry accordingly, if he can. In the case of 
the postulate of parallel lines this was done by Loba- 
chevsky and Bolyai, and their geometries are entirely 
logical. Mathematicians generally agree that the post- 


1For references, Smith, D. E., History of Modern Mathematics, 
p- 565. The best historical treatment of the subject is that by Stackel 
and Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss, 
Leipzig, 1895. 


THE BASES OF GEOMETRY 265 


ulate is not at all “self-evident.” As Klein, the well- 
known Gottingen professor, says, ‘As mathematicians 
we must array ourselves as opponents of Kant’s idea 
that the parallel axiom is to be considered an a priori 
truth.” + Lobachevsky and Bolyai postulate that through 
a given point more than one line can be drawn parallel 
to a given line, and on this, together with most of the 
axioms, postulates, and definitions of Euclid, they build 
up a perfectly consistent geometry. . 

Similarly, as in plane geometry we postulate that 
space has three dimensions and that a plane figure may 
be revolved about an axis, through three-dimensional 
space, so as to coincide with a symmetric figure, so in 
solid geometry we might postulate that a solid may be 
revolved through a four-dimensional space so as to 
coincide with a symmetric solid, e.g., a right-hand glove 
with a left-hand one. A perfectly consistent geometry 
could be constructed with this as a postulate.? 

A postulate is. not, therefore, a “self-evident”’ state- 
ment; it is a geometric assumption. In ordinary ele- 
mentary geometry we postulate only certain relations 
which most people are willing to say agree with their 
sense-perceptions. They do not entirely agree with 
them, for we have no sense-perception of a straight 


1 Vergleichende Betrachtungen, Erlangen, 1872. 

2¥For a brief and popular statement concerning the fourth dimension, 
see the recent translation of Schubert, H., Mathematical Essays and Rec- 
reations, Chicago, 1898, p. 64. 


266 THE TEACHING OF ELEMENTARY MATHEMATICS 


line, nor, a fortiort, of two parallels. Our geometric 
concepts are all abstractions made from our physical 
concepts.1 As D’Alembert says, “Geometric truths 
vare a kind of asymptote of physical truths, z.¢., the 
limit which they indefinitely approach without ever 
exactly reaching.” 

As to the number of postulates or axioms, the ques- 
tion is wholly unsettled. Practically, the teacher of 
the elements will follow those given in his text-book. 
But as has been truly said, the list usually given is 
both insufficient and superabundant, since on the one 
hand we use postulates not laid down in the ordinary 
text-books, and on the other hand we can demon- 
strate some of those which are given, so that it is 
unnecessary to assume them.? 

The most recent examination of the postulates of 
rectilinear figures is that of Hilbert,’ and is here set 
forth in some detail because of the high mathemati- 
‘cal authority with which it comes to us. “In geome- 
try we consider three different systems of things. 


1 Les figures géométriques sont de pures conceptions de l’esprit. Com- 
pagnon. 

2 De Tilly, in Rebiére, Les Mathématiques, etc., p. 21. He adds, “ The 
axioms of geometry can be reduced to three, that of distance and its 
essential properties, that of the indefinite increase of distance, and that 
of unique parallelism.” 

8 Hilbert, D., Grundlagen der Geometrie, in the Gauss-Weber-Denk- 
mals Festschrift, Leipzig, 1899. See the author’s review in The Educa- 


tional Review, January, 1900. 


THE BASES OF GEOMETRY 267 


The things of the first system we call points, desig- 
nating them A, B, C, --; the things of the second 
system we call straight lines, designating them a, 4, 
c, «+»; the things of the third system we call planes, 
designating them a, 8, y, -. The points we may 
call the elements of linear geometry; the points and 
straight lines the elements of plane geometry; the 
points, straight lines, and planes the elements of 
spatial geometry or of space. 

‘“‘We consider the points, lines, and planes in cer- 
tain mutual relations, and we designate these relations 
by the words, ‘lie,’ ‘between,’ ‘parallel,’ ‘congruent,’ 
‘continuous,’ and the exact and complete description of 
these relations follows from the axioms of geometry. 

“These axioms separate into five groups, each ex- 
pressing certain fundamental facts of our conscious- 
ness : — 

“T, Axioms of connection (Verkniipfung). 

“1. Two different points, 4, B, determine a straight 
line a, and we say that dB=a, or BA=a!} 

“2. Any two different points on a straight line de- 
fernine sthatuiere-cydie Og anded © 30 /sand 
Pas not, Comheniy Cassia, 

“3. Three non-collinear points, A, B, C, determine 
a plane a, and we say that ABC = «. 

“a4, Any three non-collinear points, A, B, C, of a 
plane «, determine «. 


1 Of course the symbol “ = ” here means “ determines,” 


268 THE TEACHING OF ELEMENTARY MATHEMATICS 


“s. If two points, A, B, of a straight line a lie in 
a plane a, then every point of a lies in «. 

“6, If two planes, a, 8, have a point 4 in common, 
they have at least one other point & in common. 

“7 In every straight line there are at least two 
points, in every plane at least three non-collinear 
points, and in space at least four non-coplanar points. 

“TTI. Axioms of arrangement (Anordnung), defining 
the concept ‘ between.’ 

“1, If A, B, C are three collinear points, and 2 lies 
between A and C, then # also lies between C and A. 

‘9, If A and C are two collinear points, there is at 
least one point B between them, and at least one point 
D such that C lies between A and D. 

“3. Of any three collinear points, there is one which 
lies uniquely between the other two. 

“a. Any four collinear points, A, B, C, D, can be so 
definitely arranged that B lies between A and C and 
also between A and LD, and that C lies between A and 
D and also between & and D. 

“x. Suppose A, &, C to be three non-collinear points, 
and a a straight line in the plane ABC, but not con- 
taining A, &, or C, if then, the straight line @ passes 
through a point within the line-segment AZ, it must 
also pass through a point within the line-segment 
BC or through a point within the line-segment AC? 

1 These five axioms of Group II were first investigated by Pasch 


(Vorlesungen iiber neuere Geometrie, Leipzig, 1882), and the fifth is 
especially due to him, 


THE BASES OF GEOMETRY 269 


“TIT. Axiom of parallelism, the denial of which 
leads to the non-Euclidean geometry. 

“TV. Axioms of congruence. 

“1. If A, B are two points on the straight line a, 
and A’ is a point on the same or another straight 
line a’, it is possible to find on a given side of a! 
from A!’ one unique point 5’ such that the line-seg- 
ment AZ (or BA) is congruent to the line-segment 
a Eee 

“2. If a line-ssegment AZ is congruent to both A’B' 
and AB", then A'S’ is also congruent to AB", 

“3, Let AB and BC be two segments of a, without 
common points; let A'S’ and B’C' be two segments 
of a’, also without common points; then if AP is 
congruent to A’B’, and BC is congruent to 4’C’, it 
must follow that AC is congruent to A’C’.” 

4. This is the axiom for angles corresponding to 
axiom 2 for segments. | 

5. This is the axiom for angles corresponding to 
axiom 3 for segments. | 

“6. If for two triangles, ABC and A’S'C’ these 
congruences exist (using ‘=’ for congruence), 


AB= A'B', AC=A'C', angle BAC =angle B'A'C’, 
then must these also exist, 
angle CBA = angle C’B'A', angle ACB = angle A'C’B’. 


“V7, Axiom of continuity (Stetigkeit)— the axiom of 
Archimedes. 





270 THE TEACHING OF ELEMENTARY MATHEMATICS 


“Let A, be any point on @ between any given 
points A and 4, suppose A,, Az, Ay ++ so taken 
that A, lies between A and A,, A, between A, and 
A,, etc., and also such that the segments 4A4,, A,A,, 
AjAg sare equal.) theme must; there. be sin the 
series: Ay, As, “4.0? as pointe Age such thatec ilies 
between A and A,.— The denial of this axiom leads 
to the non-Archimedean geometry.” 

Hilbert inserts the necessary definitions for under- 
standing these postulates (axioms), and adds numerous 
corollaries showing the far-reaching effect of the 
statements; but this is not the place to enter this 
interesting field. Whether or not his postulates are 
sufficient, it is evident that tacitly or openly they 
are assumed in our elementary rectilinear geometry. 
Their consideration should convince the teacher that 
the question of the postulates is by no means the 
simple one which the text-books sometimes make us 
feel. 

Thus geometry is exact, not because its postulates 
necessarily agree with the facts of the external 
world; that is not of so much moment. It is exact 
because it postulates definitely at the outset certain 
few statements concerning figures in space, and then 
applies logic to see what other statements can be 
deduced therefrom. 


CHAPTER XII 
TYPICAL PARTS OF GEOMETRY 


The introduction to demonstrative geometry may well 
be made independent of the text-book, unless the book 
offers some special preparatory work. If the pupils 
have not a reasonable knowledge of geometric draw- 
ing, a few days may profitably be devoted to this sub- 
ject exclusively. Professor Minchin has this to say of 
the English schools, and the same is almost as true of 
our own: ‘So far as I am able to learn by inquiry, 
Euclid is taught in all our schools without the aid 
of rule, compasses, protractor, or scale. This is quite 
in accordance with the traditional method—the classi- 
cal method which, unfortunately, so greatly domi- 
nates English education — and quite conducive to 
long-delayed knowledge of the subject. 

“Now the use of the above simple instruments for 
all beginners in geometry is the first change that I 
advocate, whether we continue to teach from Euclid’s 
book or from one proceeding on simpler and _ better 
lines. Well-drawn figures possess an enormous teach- 
ing power, not merely in geometry, but in all branches 
of mathematics and mathematical physics.” } 


1 The Teaching of Geometry, The School World, Vol. I, p. 161 (1899). 
271 


272 THE TEACHING OF ELEMENTARY MATHEMATICS 


Before undertaking the ordinary text-book demon- 
strations the teacher will also find it of great value to 
offer a few preliminary theorems which pave the 
way for the usual sequence of propositions, giving a 
notion of what is meant by a logical proof, and creat- 
ing a habit of working out independent demonstrations. 
The following, for example, might be given in this 
way: (1) All right angles are equal (if the text-book 
postulates the demonstrable fact of the equality of 
straight angles); (2) At a point in a given line not 
more than one perpendicular can be drawn to that 
line in the same plane—not that one caz be drawn, 
as so many text-books affirm but fail to prove; (3) The 
_complements of equal angles are equal; the proposi- 
tion concerning vertical angles, and several others of 
the simpler ones selected from the first ‘‘ book.” 

After a little work of this kind the pupil is prepared 
to understand the nature of a logical proof. Indepen- 
dence will begin to assert itself, confidence in his 
ability to handle a proposition without a slavish depen- 
dence upon his text-book, while mere memorizing will 
fail to secure the usual foothold at the start. These 
two points may now be impressed: (1) Every statement 
in a proof must be based upon a postulate, an axiom, a 
definition, or some proposition previously considered ; 
(2) No statement is true simply because it appears 
from the figure to be true. With this preliminary 
treatment of a dozen or more simple propositions, and 


TYPICAL PARTS OF GEOMETRY 273 


with some instruction concerning geometric drawing, 
the text-book sequence may be undertaken with much 
less danger of discouragement, of slovenly work, of 
groping in the dark, and of mere memorizing. 

Symbols — The contest between the opponents of all 
symbols and the advocates of mathematical shorthand 
in geometry, as in other branches of the science, is 
about over. In England Todhunter’s Euclid is giving 
place to the Harpur, Hall and Stevens, McKay, Nixon, 
and others which make extensive use of symbols, 
while in America Chauvenet’s excellent work has had 
to give place to less scholarly but more usable text- 
books. 

In general one is practically bound by the symbols 
in the book-in the hands of the class. A few notes 
upon the subject may, however, be suggestive. In 
the first place, only generally recognized mathematical - 
symbols should have place; in a world-subject like 
mathematics, provincialism is especially to be con- 
demned. We may think that || would be a better sign 
of equality than =, but the world does not think so, 
and we have no right to set up a new sign language. 
In this respect it is unfortunate that some of our 
American writers should continue to use the provin- 
cial symbol for equivalence (=), not only because it 
is difficult to make, but because it has no standing 
among mathematicians. Indeed, the distinction be- 


tween equal and equivalent is so nearly obliterated 
Lh 


274. THE TEACHING OF ELEMENTARY MATHEMATICS 


in our language that many teachers now use the 
more exact term ‘“‘congruent” for what some English 
writers call “identically equal,’ even though the text- 
book in their classes has the word “equal.” The 
symbol for congruence (2), a combination of the 
symbols for similarity (~, an S laid on its side, from 
similis) and equality (=), is so full of meaning and is 
so generally recognized by the mathematical world 
that its more complete introduction in elementary work 
is desirable. It is certainly not open to the objection 
of novelty, for it dates from Leibnitz, nor of the provin- 
cialism and want of significance which characterize the 
American symbol for equivalence. 

The modern symbols for limit (=, still in its provin- 
cial stage), identity (=), and non-equality (+), in addi- 
tion to the ordinary algebraic signs, are also convenient. 

There is also much advantage in following the 
modern method of reading angles and lines, and of 
lettering triangles. Among the ancients, when angles 

were always considered as less than 
i 180°, it was a matter of little moment 
whether one should read the angle 
f ; here illustrated AOB or BOA. But 

now that we recognize angles of any 
number of degrees, as when we turn a screw through 
go°, 180°, 270°, 360°, 450°, +++, it becomes necessary to 
distinguish the two conjugate angles in the figure. The 


TYPICAL PARTS OF GEOMETRY 275 


= 
QAAAS, 


ebtuse angle is, therefore, read AOS, and the reflex 
angle BOA, counter-clockwise. Pupils brought up to 
this plan from the beginning have a great advantage 
in accuracy when they come to speak of figures which 
are at all complicated. The counter-clockwise reading 
of positive angles and the clockwise reading of nega- 
tive ones is also very helpful in the generalization of 
propositions in the earlier books. 

It is also of great advantage to recognize, before 
the pupil has gone too far, the distinction between the 
line segments 48 and GA. Negative magnitudes can 
no longer be kept from elementary geometry, say what 
we may about pure form and the non-algebraic treat- 
ment of the subject. Pupils understand the negative 
magnitudes of algebra—then why not apply this 
knowledge to geometry, thus opening fields both new 
and interesting? By so doing, a mutually helpful 
correlation is established between algebra and geom- 
etry, a correlation always recognized in the more ad- 
vanced portions of the science. 

The advantage of uniformity in lettering triangles 
ABC, XYZ, +--+, in counter-clockwise order, and of 
lettering the sides opposite A, B, C, respectively, a, 4, c 
(and so for 4, y, 2, etc.), is apparent to all who have 
accustomed themselves to the arrangement. 

Reciprocal theorems — There is an interesting line of 
propositions, early met by the pupil, in which one theo- 


276 THE TEACHING OF ELEMENTARY MATHEMATICS 


rem may be formed from another by simply replacing 


the words 


point by line, 


line by potnt, 


angles of a triangle by (opposite) szdes of a triangle, 


sides of a triangle by (opposite) angles of a triangle. 


This is seen in the following propositions: 


If two triangles have 
two sides and the included 
angle of the one respec- 
tively equal to two szdes 
and the included axg/e of 
the other, the triangles are 
congruent. 

If two szdes of a triangle 
are equal, the anxgles oppo- 
site those szdes are equal. 


If two triangles have 
two angles and the includ- 
ed szde of the one respec- 
tively equal to two angles 
and the included szde of 
the other, the triangles are 
congruent. 

If two angles of a triangle 
are equal, the szdes oppo- 
site those angles are equal. 


Of course the teacher may pass over this relation- 
But 
there is great advantage in recognizing the parallelism 


ship, as most text-books do, without comment. 
early in the course, for two reasons: (1) It adds greatly 
to the pupil’s interest to see this symmetry of the sub- 
ject, to note that certain propositions have a dual; 
and (2) It often suggests new possible theorems for 
investigation —the pupil has the interest of discov- 
ery. This is seen in the following simple exercise: In 
a triangle ABC, where a= 4, the bisector of angle C, 


TYPICAL PARTS OF GEOMETRY 277 


produced to ¢, bisects sede c. The pupil who is led to 
discover the reciprocal theorem, and to investigate its 
validity (for reciprocal statements are not always true), 
has opened before him a field of perpetual interest, a 
field in which he is an independent worker. 

Converse theorems are often thought uninteresting. 
Students get the idea that the converses are always 
true, and that it is a stupid waste of time to prove them. 
And yet, so necessary are these propositions to the 
logical sequence of geometry, that they have an impor- 
tant place. In arranging to present the subject to a 
class, the teacher is met by three questions: (1) What 
are converse theorems? (2) Are converses always true? 
(3) How are converse theorems best proved? 

Two theorems are said to be converse, each of the 
other, when what is given in the one is what is to be 
proved in the other, and wice versa. For example, “In 
triangle ABC, if a= 0 then angle A = angle BS,” and, 
instriancle ABC if angle A=angle 4 then a=," 
are converses, and each is true; but if the second one 
should read, “In triangle AAC, if all the angles are equal 
then a = 4,” the two would not be converses, although 
what is given in the first (a = 0) is what is to be proved 
in the second, for the vice versa element is wanting. 

The class should be made aware of numerous false 
converses, that the necessity for proof may be appreci- 
ated. For example, “All right angles are equal angles,” 
“Tf a triangle contains a right angle it is not an equi- 


278 THE TEACHING OF ELEMENTARY MATHEMATICS 


lateral triangle,” “If two numbers are prime their 
product is composite,” are all true statements, but their 
converses are not. 

There are so many converses to be proved that the 
teacher will find it advantageous, both as to time and 
logic, to consider the Law of Converse rather early in 
the course. At the expense of one or two lessons 
given to the understanding of the law, the time should 
be spared, since it will be amply repaid later. The law 


is as follows: 


Whenever three theorems have the following relations, 


their converses must be true: 


1. If it has been proved that when A>8, then X> Y, and 
2. If it has been proved that when A= 4S, then X= Y, and 
3. If it has been proved that when 4 < JS, then X< Y, 
then the converse of each is true. For 

If X> Y, then A can neither be equal to nor less 
than & without violating 2 or 3; .. 4 >8, which 
proves the converse of I. 

lf X= Y, then A can, neither be. greater norless 
than #8 without: violating ror 3; .. A= 2B; which 
proves the converse of 2. 

If X< Y, then A can neither be greater than nor 
equal to & without violating 1 or 2; .. A < B&B, which 
proves the converse of 3. 

This law, proved once for all, enables us to prove 
such of the converses as we need in elementary geom- 


TYPICAL PARTS OF GEOMETRY 279 


etry without using the tedious demonstration of Euclid 
with every case. For example, as soon as it has been 
proved that, in triangle ABC, if A= B then a=4, and 
if A>B then a>d (which, by mere change of letters 
in the figure, also proves that if d<# then a< 3d), 
this law shows that the three converses are true. 
Should any teacher feel that this is too difficult for 
beginners, it should be noticed that the proof is iden- 
tical with that usually given, but it is here merely 
set forth for subsequent use, and is given a name. 
Generalization of figures — Until recently elementary 
geometry seemed afraid to consider a reflex angle, or 
a concave polygon, or an equilateral triangle as a 
special case of an isosceles triangle, to say nothing 
of a cross polygon, or a cylinder with a non-circular 
directrix, or a negative line-ssegment. But our best 
recent works have presented these and other modern 
ideas in such a simple fashion that their general in- 
troduction cannot long be delayed. It is not at all 
a matter of the text-book; it lies with the teacher to 
make much or little of it, and scarcely any feature 
of the work adds more interest, develops more orig- 
inality, or better paves the way for future progress. 
Take the familiar theorem that the sum of the interior 
angles of an m-gon equals “— 2 straight angles, 
stated, of course, in various ways and with more or 
less circumlocution. After it has been proved for 
the simple convex figure, the teacher may ask if it 


"260 .LHE TEACHING OF ELEMENTARY MATHEMATICS 


is true in case one angle becomes reflex; he may 
then move the vertex back until the angle becomes 
straight, and ask the same question. Students have 
no trouble with such questions, and they readily 
follow a teacher to the consideration of the cross 
polygon, a case best presented by moving the vertex 
of a marked angle through one of the opposite sides. 

The case of the sum of the exterior angles of a 
polygon is also a valuable one for beginners. If the 
student will letter the angles for the ordinary convex 
polygon, and keep the same lettering when it becomes 
concave or cross, he will find that the proof is the same 
for all cases. When the angle AOS, for example 
(always read counter-clockwise), becomes GOA, it is to 
be considered negative, but otherwise the proof is quite 
unchanged. Indeed, the one (practically unvarying) 
principle to be given the student is this: Letter the sim- 
ple figure properly, keeping the same letters in all trans- 
formations, and the proof will be the same for all cases. 

The principle is well illustrated in the case of the 
square on the side opposite an obtuse angle of a 
triangle. It equals the sum of the squares on the 
other sides p/us twice acertain rectangle. As the angle 
becomes less obtuse this rectangle becomes smaller; if 
the angle becomes right, this rectangle vanishes and 
the theorem becomes the Pythagorean ; if the angle 
becomes acute, a certain projection becomes negative, 
making the rectangle negative, and instead of having 


TYPICAL PARTS OF GEOMETRY 281 


plus twice a certain rectangle we have mznus twice 
that rectangle, the proposition becoming the one con- 
cerning the square on the side opposite an acute angle. 

This generalization of typical figures materially 
lessens the detail of geometry. For example, the 
propositions concerning the measure of an inscribed 
angle, an angle formed by a tangent and a chord, an 
angle formed by two chords, or two secants, or a secant 
and a tangent, or two tangents, are all special cases 
of a single theorem. It would be unwise to give this 
general theorem first, but after considering the cases 
of an inscribed angle, and the angle formed by a chord 
and tangent, classes have no trouble in taking the gen- 
eral case and in so transforming the figure as easily to 
get the special cases from it. The proof has only a 
couple of steps in the most general form, and it is a 
waste of time to make special theorems for each of the 
various simple cases. 

The proposition concerning the “product” of the 
segments of two intersecting chords, or secants, is also 
one which is often extended through three or four 
theorems. It requires only two steps to prove the 
general case. Ifa pencil of lines cuts a circumference, 
the rectangle (product) of the two segments from the 


1 Upon this set of theorems, however, the teacher should read the 
report of the sub-committee on mathematics in the Report of the Com- 
mittee of Ten, Bulletin No. 205 of the U.S. Bureau of Education, p. 113. 


The position there taken is, however, open to very serious question. 


282 THE TEACHING OF ELEMENTARY MATHEMATICS 


vertex is constant whichever line is taken. From this 
theorem four or five others come as special cases by 
simply transforming the figure slightly. The time has 
surely passed for fearing so valuable a phrase as “ pencil 
of lines.” 

These few illustrations suffice to show that elemen- 
tary geometry offers a field, interesting to teachers and 
pupils alike, for simple generalizations. The danger 
lies on the one side in always attempting to give the 
general before the particular (a fatal error), and on the 
other in cutting out all of the interest which comes 
from generalization, thus falling into the old humdrum 
of multiplying theorems to fit all special cases. 

Loci of points — Most of our elementary works devote 
some space to the treatment of a few simple loci of 
points, the reciprocal subject of “sets of lines” being 
generally regarded as hardly worth considering at this 
stage of the student’s progress. The subject is of 
little or of great value, depending on the use subse- 
quently made of it. A few of our recent text-books 
have carefully explained the term “locus,” and have 
given satisfactory proofs of the theorems, but the 
majority fail in two particulars, and as to these a few 
words may be of value. 

To say that the locus of points (in a plane) is the 
line containing those points, is entirely inadequate, 
for this line may contain other points, or the locus 
may consist of two or more lines, or of a line and a 


TYPICAL PARTS OF GEOMETRY 283 


point (as in the locus of a point 7 distant from a 
circumference). Perhaps the best plan is to fall back 
on the etymology of Jocus (Lat. place) and say, The 
place of all points satisfying a given condition is 
called the locus of points satisfying that condition — 
giving further explanation by means of illustration. 

But the most serious error usually found is in the 
proof. ‘In proving a theorem concerning the locus of 
points it is necessary and sufficient to prove two things: 
(1) That any point on the supposed locus satisfies the 
condition ; (2) That any point not on the supposed locus 
does not satisfy the condition. For if only the first 
point were proved, there might be some other line in 
the locus; and if only the second were proved, the sup- 
posed locus might not be the correct one.” <A text-book 
which fails in these points should be discarded. 

Methods of attack— There is a certain value in 
turning a pupil into a chemical laboratory, after he 
has seen some experiments performed, and there 
telling him to discover something new, or to find the 
atomic weight of some substance. He will fail, but 
the attempt may serve to broaden his ideas. It is 
also of some value to hand him a few crystals, tell- 
ing him to prove that they are this or that kind of 
salt, leaving him to his own devices. But the teacher 
who would do this with elementary students, who 
would offer no general directions as to methods of 


attack, who would allow a student to wander aim- 


284 THE TEACHING OF ELEMENTARY MATHEMATICS 


lessly about, groping blindly and wasting his energies 
in futile attempts, would be looked upon as a failure. 
And yet this is about what we usually find in a class 
in geometry; students are turned loose among a mass 
of exercises, and are told to invent new proofs, to 
find new theorems, to solve problems and prove theo- 
rems entirely new tothem. Their only hint is that given 
by the demonstration of some recent proposition ; their 
only hope, to stumble upon the proof—to draw the 
prize ticket in the lottery without too great delay. 

Mathematicians do not proceed in any such way; 
they call to their assistance all the general methods 
possible, and to the teacher of geometry this should 
be a lesson. The discovery of theorems, new at 
least to the pupil and probably to the teacher, is an 
interesting application of the law of reciprocity 
already mentioned. Thus if a student knows Pascal’s 
“mystic hexagram” (If the opposite sides of an in- 
scribed hexagon intersect, they determine three col- 
linear points), it is but a step to rediscover, in the 
same way that it was originally found, Brianchon’s 
well-known theorem.} 


1 The teacher will find this theory worked out fully in Henrici and 
Treutlein’s Lehrbuch der Elementar-Geometrie, Leipzig, 1881, 3. Aufl, 
1897, one of the most suggestive works on the subject. An excellent 
little handbook which deserves a place in the library of every teacher 
of elementary mathematics is Henrici’s Elementary Geometry, Congruent 
Figures, London, 1879, — a work in which the reciprocity idea is brought 
out quite fully. 


TYPICAL PARTS OF GEOMETRY 285 


But it is to methods of attack in the treatment of 
exercises that it is desired to direct especial attention. 
This subject has received much consideration at the 
hands of Petersen,! Rouché and De Comberousse,? and 
Hadamard,® and the following suggestions are largely 
from their works. 

1. In attacking a theorem take the most general 
figure possible. £&.g., if a theorem relates to a tri- 
angle, draw a scalene triangle; one which is equi- 
lateral or isosceles often deceives the eye and leads 
away from the demonstration. 

2. Draw all figures as accurately as possible. An 
accurate figure often suggests a demonstration. On 
the other hand, the student who relies too much upon 
the accuracy of the figure in the demonstration is 
liable to be deceived. 

3. Be sure that what is given and what is to be 
proved are clearly stated with reference to the letters 
of the figure. Neglect in this respect is a fruitful 
cause of failure. 

4. Then begin by assuming the theorem true; see 
what follows from that assumption; then see if this 


1 Methods and Theories of Elementary Geometry, London and Copen- 
hagen, 1879. 

2 Traité de Géométrie, 6 éd., Paris, 1891. 

8 Lecons de Géométrie élémentaire, Paris, 1898. 

4 The immediate source is, however, Beman and Smith’s New Plane and 
Solid Geometry, Boston, 1899, p. 35, 152, to which reference is made for 
further details. 


286 THE TEACHING OF ELEMENTARY MATHEMATICS 


can be proved true without the assumption; if so, try 
to reverse the process. 

5. Or begin by assuming the theorem false, and 
endeavor to show the absurdity of the assumption 
(veductio ad absurdum). 

6. To secure a clearer understanding of the propo- 
sition to be proved it is often well to follow Pascal’s 
advice, and “substitute the definition in place of 
the name of the thing defined.” 

7. In attempting the solution of a problem the 
method of analysis suggested in 4, above, will usually 
lead to success. Assume the problem solved, con- 
sider what results follow, and continue to trace these 
results until a known proposition is reached; then 
seek to reverse the process. 

8. One of the most fruitful methods of attacking 
problems is by means of the intersection of loci. So 
long as it is known merely that a point is on ove line, 
its position is not definitely determined; but if it is 
known that the point is also on another line, its posi- 
tion may (and if both lines are straight must) be 
uniquely determined. For example, if it is known 
that a point is on a certain straight line and a certain 
circumference, it may be either of the two points of 
intersection. Thus, in a plane, to find a point equally 
distant from two fixed points, 4, &, and also equally 
distant from two fixed intersecting lines, x, vy; the 
locus of points equidistant from 4 and J is the per- 


TYPICAL PARTS OF GEOMETRY 287 


pendicular bisector of AB; the locus of points equi- 
distant from 2 and y is the pair of lines bisecting the 
angles ry and yr; since, in general, the first line 
will cut the other two in two points, both of these 
points answer the conditions. 

Petersen gives numerous other methods, but the 
above suggestions answer very well for all cases the 
student will meet in elementary geometry. 

Ratio and proportion—In the treatment of this 
chapter we have two extremes of method. First 
there is the Euclidean, purely geometric, scientific 
and logical to the extreme. It is because of this 
treatment that English teachers sometimes argue the 
more strongly for Euclid — although in practice they 
never use the chapter! The fact is, it is altogether 
too difficult for beginners, even as modified by the 
syllabus of the Association for the Improvement of 
Geometrical Teaching. One has but to read the 
Euclidean definition of equal ratios to be assured of 
this fact: “The first of four magnitudes is said to 
have the same ratio to the second, which the third 
has to the fourth, when any equimultiples whatsoever 
of the first and third being taken, and any equi- 
multiples whatsoever of the second and fourth; if the 
multiple of the first be less than that of the second, | 
the multiple of the third is also less than that of the 
fourth: or, if the multiple of the first be equal to 
that of the second, the multiple of the third is also 


288 THE TEACHING OF ELEMENTARY MATHEMATICS 


equal to that of the fourth; or, if the multiple of 
the first be greater than that of the second, the 
multiple of the third is also greater than that of the 
Tourth.s 

The other extreme is the purely algebraic plan, the 
one adopted by most American text-book writers, a 
plan entirely non-geometric, unscientific, a break in 
the logic of geometry, but so easy that neither teacher 
nor pupil need.do much serious thinking to master it. 
Occasionally a writer inserts a proposition at the end 
of the chapter, intending to bridge the chasm between 
algebra and geometry, but it rarely creates any im- 
pression upon the student. 

Between these extremes, the strictly scientific and 
the strictly unscientific, the too difficult and the too 
easy, the geometric and the algebraic, the serious 
and the trivial, there is at least one fairly scientific 
and usable mean. It consists in proving that there 
is a one-to-one correspondence between algebra and 
geometry, with this relationship : 


Geometry. Algebra. 
A line-segment. A number. 
The rectangle of two line- The product of two numbers. 


segments. 


This having been made a matter of proof, it is 
further postulated that any geometric magnitude may 


1 Blakelock’s Simson’s Euclid, London, 1856. 


TYPICAL PARTS OF GEOMETRY 289 


be represented by a number. With these assump- 
tions and proofs, the laws of proportion may be 
proved either by algebra or by geometry, as may 
be the most convenient. The first proposition, stated 
in dual form, would then read: 


If four xumbers are in If four /zzes are in pro- 
proportion, the product of portion, the rectangle of 
the means equals the grod- the means equals the vec- 
uct of the extremes. tangle of the extremes. 


The impossible in geometry— While it does not 
enter into the province of the teacher to require the 
pupil to attempt the impossible, at the same time 
the questions of the limits of the possible frequently 
arise even in plane geometry. 

To say that nothing is impossible, is to make a 
pleasant sounding epigram, and if it means that it is 
possible, given infinite power, to do any particular 
thing, it is true. It merely asserts that nothing is 
impossible if one has the means to insure its possi- 
bility. But the moment that limitations are imposed, 
the epigram ceases to be true. To draw a circle 
with the compasses is possible; with the straight- 
edge only, it is impossible. To draw a straight line is 
possible, but if one is limited to the use of the com- 
passes it becomes impossible. To draw an ellipse, 
hyperbola, cissoid, or conchoid,—all these are pos- 
sible zf the necessary instruments are allowed, but 

U 


290 THE TEACHING OF ELEMENTARY MATHEMATICS 


they are impossible with simply the compasses and 
straight-edge. 

From remote antiquity men have tried to trisect 
an angle, a problem simple enough if the necessary 
instruments are allowed, but one well known by 
mathematicians to have been proved to be impossible 
by the use of compasses and straight-edge alone. 
It is not that the world has not yet solved it, be- 
cause, like the fact that #*+ 4” cannot equal 2” for 
Ae 2 ait might sometimes yield to proof; but it has 
already been proved that it cannot be solved. 

Similarly the problem of constructing a square 
equal to a given circle, “squaring the circle,” is easy 
enough if one may use a certain curve, but it has 
been proved to be impossible by the use of the 
instruments of elementary geometry. In the same 
category belong the problems of the duplication of 
the cube, and the construction of the regular hepta- 
gon. The world is full of circle-squarers, and cube- 
duplicators, and angle-trisectors, simply because these 
elementary historic facts are unknown. 

Solid geometry — Euclid paid little attention to solid 
geometry, with the result that his followers in the 
English schools have also neglected it. Since the con- 
servative Eastern states have always been influenced by 


1 Upon this and other problems mentioned in this connection, the most 
accessible work for teachers is Klein’s Famous Problems of Elementary 
Geometry, English, by Beman and Smith, Boston, 1896, 


TYPICAL PARTS OF GEOMETRY 291 


the educational traditions of England, solid geometry has 
never had the hold in the preparatory schools that it 
has in the Central and Western states, where tradition 
counts for less. The argument on the one side is this: 
In the time at our disposal we cannot teach all of plane 
geometry, to say nothing of the solid—as if all of 
plane geometry could ever be taught! The argument 
on the other side is this: The whole question is one of 
degree; with a year at the teacher’s disposal, he would 
do better to teach plane geometry about two-thirds of 
the time, and solid geometry one-third; this would give 
mental training at least equally valuable, which is the 
first consideration, it would add to the pupil’s interest, 
and it would contribute to the practical side through 
the added knowledge of mensuration. 

The effort has several times been made to work out a 
feasible plan for carrying solid geometry along side by 
side with the plane.| The scheme has a number of 
advantages. It is interesting, for example, to pass a 
plane through certain solids (to slice into them, so to 
speak), and get figures of plane geometry out of them. 
It is also interesting to note the one-to-one correspond- 
ence between the spherical triangle, the trihedral angle, 
and the plane triangle. But while, theoretically, this 
scheme is quite feasible, practically it has few followers. 
It is contrary not only to the historical development of 
the science, but also to psychology; it makes the com- 


1 F.g., Paolis, R. de, Elementi di Geometria, Torino, 1884. 


292 THE TEACHING OF ELEMENTARY MATHEMATICS 


plex contemporary with the simple, the general with the 
particular, from the very first. It is interesting, how- 
ever, to see how skilfully the Italian writers are han- 
dling the matter. 

Practically, it has been found best to take up the 
demonstrative solid geometry after a course in plane 
geometry has been completed. The subject then offers 
few difficulties to most students; a little patience at the 
outset, a few simple pasteboard models, easily made by 
the class, care in drawing the first figures so as to bring 
out the perspective, — these are the considerations nec- 
essary in beginning work in the geometry of three 
dimensions. Models, preferably to be made by the 
student, are crutches to be used until the beginner can 
walk, then to be discarded. To keep them, to have 
special ones for every proposition, éven to have their 
photographs, is to take away one of the very things 
we wish to cultivate,—the imagination, the power.of 
imaging the solids, the power of abstraction. In gen- 
eral, the appeal to models should be made only as it 
is necessary to return to the crutch— when the pupil 
falters. 

The same is true of the spherical blackboard; it is 
valuable and should be used in every school, especially 
in the consideration of polar and symmetric triangles ; 
but never to depart from it in spherical geometry, or 
never to take up a theorem without a photograph of 
the sphere, is wholly unwarranted by necessity or by 


TYPICAL PARTS OF GEOMETRY 293 


the demands of education. The student needs to make 
abstractions, to get along with a figure drawn on a plane, 
and to be able to work independent of the sphere or 
its photograph. 

The teacher will do well to add to the treatment 
usually given some little discussion of recent features for 
which we are indebted to the Germans. A consider- 
able saving is effected in “producing” lines, planes, and 
curved surfaces, in treating prisms, pyramids, cylinders, 
and cones, by the introduction of the notion of pris- 
matic, pyramidal, cylindrical, and conical surfaces and 
spaces. The concepts are simple, and by their use a 
number of proofs are considerably shortened. The 
prismatoid formula, introduced by a German, E. F. 
August, in 1849, should also have place on account of 
its great value in mensuration. Euler’s theorem, which 
states that in the case of a convex polyhedron with 
e edges, uw vertices, and f faces, e+2=f+y, also 
deserves place, both for the reasoning involved and 
its interesting application to crystallography. These 
additions are easily made, whatever text-book is in 
use, and teachers will find them of great value. The 
objection on the score of difficulty is groundless. 

The one-to-one correspondence between the poly- 
hedral angle and the spherical polygon should also be 
noted, a correspondence not always sufficiently prom- 
inent in our text-books. This relation may be set forth 
as follows: 


294 THE TEACHING OF ELEMENTARY MATHEMATICS 


‘Since the dihedral angles of the polyhedral angles 
have the same numerical measures as the angles of the 
spherical polygons, and the face angles of the former 
have the same numerical measure as the sides of the 
latter, it is evident that to each property of a polyhedral 
angle corresponds a reciprocal property of a spherical 
polygon, and vice versa. This relation appears by 
making the following substitutions: 


Polyhedral Angles. Spherical Polygons. 
a. Vertex. a. Centre of Sphere. 

b. Edges. 6. Vertices of Polygon. 
c. Dihedral Angles. c. Angles of Polygon. 
d. Face Angles. ad. Sides. 


“In addition to the correspondence between polyhe- 
dral angles and spherical polygons, it will be observed 
that a relation exists between a straight line in a plane 
and a great-circle arc on a sphere. Thus, to a plane 
triangle corresponds a spherical triangle, to a straight 
line perpendicular to a straight line corresponds a great- 
circle arc perpendicular to a great-circle arc, etc.” It 
may also be mentioned, in passing, that the word “arc” 
is always understood to mean “great-circle arc,” in the 
geometry of the sphere, unless the contrary is stated. 

A further relationship of interest in the study of 
solid geometry is that existing between the circle 
and the sphere, and illustrated in the following state- 
ments : 


TYPICAL PARTS OF GEOMETRY 


“The Circle. 


A portion of a Ze cut off by 
a circumference is a chord. 

The greater a chord, the less 
its distance from the centre. 

A diameter (great chord) bi- 
sects the cycle and the circum- 
ference. 

Two diameters (great chords) 
bisect each other. 


295 


The Sphere. 


A portion of a plane cut off by 
a spherical surface is a circle. 

The greater a czrc/le, the less 
its distance from the centre. 

A great circle bisects the 
sphere and the spherical sur- 
face. 

Two great circles bisect each 
other. 


Hence may be anticipated a line of theorems on the sphere, 
derived from those on the circle, by making the following substi- 


tutions: 


Ton Cercle, 12... cy cums erence, 
3. line, 4. chord, 5. diameter. 


1. Sphere, 2. spherical surface, 
3. Blane, 4. circle, 5. great circle.” 


The advantage in noticing this one-to-one correspond- 
ence is evident if we consider some of the theorems. 
In the following, for example, a single proof suffices 


for two propositions: 


If a ¢trihedral angle has 
two dihedral angles equal 
to each other, the opposite 
face angles are equal. 


If a spherical triangle has 
two angles equal to each 
other, the opposite szdes 
are equal. 


The generalization of figures already mentioned in 


speaking of plane geometry here admits of even more 
extended use. It is entirely safe to take up the men- 
suration of the volume or the lateral area of the frus- 
tum of a right pyramid, and then let the upper base 


shrink to zero, thus getting the case of the pyramid 


296 THE TEACHING OF ELEMENTARY MATHEMATICS 


as a corollary, or let it increase until it equals the lower 
base, thus getting the case of the prism; the prism 
would, however, naturally precede the frustum. So for 
the frustum of the right circular cone, and the cone and 
cylinder, a method not only valuable from the consider- 
ation of time, but also for the idea which it gives of the 
transformation of figures. 

Most of these suggestions can be used to advantage 
with any text-book. Some are doubtless used already 
by many teachers, and it is hoped all may be of value. 


CHAPTER XIII 
THE TEACHER’S BOOK-SHELF 


Although in this work considerable attention has 
already been paid to the bibliography of the subject, 
a few suggestions as to forming the nucleus of a 
library upon the teaching of mathematics may be of 
value. It has been the author’s privilege, after lecturing 
before various educational gatherings, to reply to many 
letters asking for advice in this matter, and so he 
feels that there are many among the younger genera- 
tion of teachers who will welcome a few suggestions 
in this line. 

In the first place, the accumulation of a large num- 
ber of elementary text-books is of little value. The 
inspiration which the teacher desires is not to be 
found in such a library; such inspiration comes rather 
from a few masterpieces. Twenty good books are 
worth far more than ten times that number of ordi- 
nary text-books. Hence, in general, a teacher will 
do well never to buy a book of the grade which he 
is using with his class; let the book be one which 
shall urge him forward, not one which shall make 
him satisfied with the mediocre. 

297 


298 THE TEACHING OF ELEMENTARY MATHEMATICS 


Since an increasing number of teachers, especially 
in our high schools, have some knowledge of German 
or French, and would be glad to make some use of 
that knowledge if encouraged to do so, it should be 
said that the best works which we have upon general 
methods of attacking the various branches of mathe- 
matics are in French. The best works, as a whole, 
illustrating progress in particular branches, are in 
German, although some excellent works in special 
_ lines are to be found in Italian. The other Contt- 
nental languages offer but little of value that has not 
been translated into English, French, or German. 

Arithmetic — The teacher of primary arithmetic 
needs to consult works on the science of educa- 
tion rather than those upon the subject itself, both 
because all of our special writers seem to hold a brief © 
for some particular device, and because the mathe- 
matical phase of the question is exceedingly limited. 
DeGarmo’s Essentials of Method (Boston, Heath) and 
the McMurrys’ General Method and their Method of 
the Recitation (Bloomington, Public Sch. Pub. Co.) are 
among the best American works. Along the special 
line, for teachers who will guard against going to 
extremes, may be recommended Grube’s Leitfaden 
(translated by Levi Seeley, New York, Kellogg, and 
by F. Louis Soldan, Chicago, Interstate Pub. Co.), 
Hoose’s Pestalozzian Arithmetic (Syracuse, Bardeen), 
Speer’s New Arithmetic (Boston, Ginn), and Phillips’s 


THE TEACHER’S BOOK-SHELF 301 


German bibliography of the several subjects. Al- 
though advocating a particular method, and therefore 
outside of the general province of this bibliography, 
mention should be made of Knilling’s latest work, 
Die naturgemasse Methode des Rechenunterrichts 
in der deutschen Volksschule (Miinchen, Olden- 
bourg), on account of its psychological review of the 
problem of elementary arithmetic. 

Algebra — One of the first works which a teacher 
may profitably own is Chrystal’s Algebra (two vol- 
umes, New York, Macmillan), a work which he will not 
soon master, but a fountain from which he will get 
continual inspiration. Since this enters but little into 
the subject of the equation, it should be supplemented 
by Burnside and Panton’s Theory of Equations (Dub- 
lin, Hodges). To these may well be added that 
multum in parvo, Fine’s Number System of Algebra 
(Boston, Leach). 

The most scholarly elementary algebra that has 
appeared in recent years is Bourlet’s Algébre élémen- 
taire (Paris, Colin), a work which is thoroughly up 
to date and which contains a large amount of new 
matter which is usable in high-school work. Of 
course there are many other excellent algebras in 
French, some of them much more extensive than 
Bourlet, but none can be so highly recommended as 
the first work to be purchased. 

From the standpoint of method, especially as ap- 





302 THE TEACHING OF ELEMENTARY MATHEMATICS 


plied to the earlier stages, Schiiller’s Arithmetik und 
Algebra (Leipzig, Teubner) deserves a place. It is a 
practical book by a practical teacher. German works, 
however, run off into special lines to such an extent 
that it becomes difficult to select a small number. 
For the teacher who is taking classes through literal 
equations, and who wishes to somewhat master the 
subject, Matthiessen’s Grundziige der antiken und 
modernen Algebra der litteralen Gleichungen (Leip- 
zig, Teubner) will prove a gold mine, but it is not at 
all of the nature of a text-book. Quite a remarkable 
little work, condensing the modern theory of equa- 
tions in small compass, is Petersen’s Theorie der 
algebraischen Gleichungen (Kopenhagen, Host). If 
one cares to look into higher algebra,,Weber’s Lehr- 
buch der Algebra (two volumes, Braunschweig ; 
Vol. I, French by Griess, Paris, Gauthier-Villars), 
or Biermann’s Elemente der hohere Mathematik 
(Leipzig, Teubner), are the best of the recent works. 
There are also a few recent, scholarly, and inexpen- 
sive works published in the Sammlung Géschen and 
the Sammlung Schubert which will prove of value 
out of all proportion to the cost. (See p. 176, note.) 

Geometry — The teacher of geometry should have 
some good edition of Euclid. On account of its second 
-volume on solid geometry (Geometry in Space, Oxford, 
Clarendon Press), Nixon’s may be recommended, 
although the Harpur Euclid, Hall and Stevens (New 


THE TEACHER’S BOOK-SHELF 299 


article in the Pedagogical Seminary for October, 1897. 
But the most scholarly work upon this subject that 
America has produced is McLellan and Dewey’s Psy- 
chology of Number (New York, Appleton), a work 
which the author believes to go somewhat to an extreme 
in its ratio idea, but one which every teacher should 
place upon his shelves and frequently consult. 

Along higher lines, Brooks’s Philosophy of Arith- 
metic (Philadelphia, Sower) deserves a place. Its his- 
torical chapter is unreliable, and it runs too much 
to cases, rules, and formulae, but it has many good 
features, and it is worthy of recommendation. As 
showing the views of recent educators as to what mat- 
ter should be eliminated, what new subjects should 
be added, and how the leading topics may be treated, 
the author ventures to suggest Beman and Smith’s 
Higher Arithmetic (Boston, Ginn). 

In French there is little of value upon primary 
arithmetic. Upon higher arithmetic, however, numer- 
ous works have appeared which cannot fail to inspire 
the ‘teacher. Of these the best is Jules Tannery's 
Lecgons d’Arithmétique théorique et pratique (Paris, 
Colin), although Humbert’s Traiteé d’Arithmétique 
(Paris, Nony) is also a valuable work. For one who 
cares to go into the theory of numbers there is no 
better introduction than Lucas’s Théorie des Nom- 
bres (Paris, tome 1, Gauthier-Villars). 

In German there is a veritable emdbarras de richesses. 


300 THE TEACHING OF ELEMENTARY MATHEMATICS 


The number of works upon primary arithmetic, and of 
text-books designed to carry out particular schemes, is 
appallingly great. It would be unwise for one begin- 
ning a library to attempt to purchase this class of 
works. It is better to put upon the shelves a few 
works which weigh these various methods, presenting 
their distinguishing features in brief compass. The 
best single work to purchase is Unger’s Die Methodik 
der praktischen Arithmetik in historischer Entwickel- 
ung (Leipzig, Teubner), the latter part of which sets 
forth the characteristics of the plans suggested by 
Pestalozzi, Tillich, Stephani, Von Tiirk, Diesterweg, 
Grube, Tanck, Knilling, e¢ al. A second work of 
great value is Janicke’s Geschichte der Methodik des 
Rechenunterrichts, which, with Schurig’s Geschichte 
der Methode in der Raumlehre, forms the third 
volume of Kehr’s Geschichte der Methodik des Volks- 
schulunterrichtes (Gotha, Thienemann), but which may 
be purchased separately. A third work, hardly up to 
those mentioned, however, is Sterner’s Geschichte der 
Rechenkunst (Miinchen, Oldenbourg), the latter part 
of which is devoted to comparative method. For the 
most scholarly treatment of arithmetic, elementary 
algebra, and elementary geometry, as of other sub- 
jects, by grades, the teacher should own a copy of 
Rein, Pickel and Scheller’s Theorie und Praxis des 
Volksschulunterrichts nach Herbartischen Grundsatzen 
(Leipzig, Bredt), a work which also sets forth the 


THE TEACHER’S BOOK-SHELF 303 


York, Macmillan), and others, are excellent. As an 
introduction to the recent development of elementary 
geometry, Casey’s Sequel to Euclid (Dublin, Hodges) 
should be among the earliest purchases, and to this may 
also be added, with profit, three recent manuals by 
M’Clelland (Geometry of the Circle, Macmillan), Du- 
puis (Synthetic Geometry, Macmillan), and Henrici 
(Congruent Figures, London, Longmans). 

In France, where they are not tied to Euclid, nor 
even to Legendre, there is more flexibility in the course 
than is found in England. Accordingly the modern 
notions of geometry have more readily found place, and 
the reader of French will find some very inspiring 
literature awaiting him. Probably the best single work 
for the teacher of geometry, in any language, is Rouché 
and De Comberousse’s Traité de Géométrie (Paris, 
Gauthier-Villars). Of the recent works, Hadamard’s 
Lecons de Géométrie élémentaire (Paris, Colin) is one 
of the best. 

In Germany still more flexibility is shown than in 
France. The making of geometry an exercise in logic, 
which England carries to an extreme, and which Amer- 
ica and France possibly carry too far, is not so notice- 
able in Germany. The result is a shorter course, one 
divested as far as possible of propositions in the nature 
of lemmas, but one in which modern ideas find wel- 
come. To appreciate this spirit the teacher should 
purchase Henrici and Treutlein’s Lehrbuch der Ele- 


304 THE TEACHING OF ELEMENTARY MATHEMATICS 


mentar-Geometrie (Leipzig, Teubner), one of the best 
books published. As a type of the best of the inex- 
pensive handbooks, it would be well to add Mahler’s 
Ebene Geometrie (Sammlung Géoschen, Leipzig, — it 
costs but twenty cents in Germany), a bit of concen- 
trated inspiration. 

Italy has produced some excellent works on element- 
ary geometry; indeed, in some features, it has been 
the leader. Socci and Tolomei’s Elementi d’ Euclide 
(Firenze, 1899), Lazzeri and Bassani’s Elementi di 
Geometria (Livorno, 1898), Faifofer’s various works 
(Venezia, Tipog. Emiliana), and Paolis’s Elementi di 
Geometria (Torino, Loescher), all have distinguishing 
features which would entitle them to a place upon the 
shelves of the reader of Italian. 

History and general method — Probably the most 
practical works on mathematical history to purchase at 
first are Ball’s (Macmillan) and Fink’s (Beman and 
Smith’s translation, Chicago, Open Court). The former 
is the more popular, the latter the more mathematical. 
Cajori has also written two readable works upon the 
general subject (Macmillan). The leading works are, 
however, in German, and have been mentioned in the 
foot-notes. 

On general method the pioneer among prominent 
writers was Duhamel, whose Des Méthodes dans les 
Sciences de Raisonnement (Paris, Gauthier-Villars) fills 
five volumes. The work is not, however, of greatest 


THE TEACHER’S BOOK-SHELF 305 


practical value to the teacher of to-day. Dauge’s 
Cours de Méthodologie mathématique (Paris, Gauthier- 
Villars) is comparatively recent, but this, too, fails to 
touch the vital points in which the elementary teacher 
is especially interested. Laisant’s La Mathématique 
(Paris, Carré et Naud), frequently mentioned in this 
work, is a small volume, but it is one of the best efforts 
of its kind, and it may well have a place upon the 
teacher’s book-shelf. Clifford’s Common Sense of the 
Exact Sciences (Appleton) should also be at hand for 
consultation. 

In the way of periodical literature, Enestrém’s Bib- 
liotheca Mathematica (Leipzig, Teubner) is one of the 
best publications devoted to the history of the subject. 
As to general mathematical teaching, Hoffmann’s Zeit- 
schrift fiir mathematischen und naturwissenschaftlichen 
Unterrichts (Leipzig, Teubner), and L’Enseignement 
Mathématique, Revue Internationale (bi-monthly, Paris, 
Carré et Naud), are among the best. 





INDEX 


[Of several foot-note references to the same work, only the first is given.] 


Aahmesu. See Ahmes. 
Abacus, 57, Iot. 
Eneas Sylvius, 13. 
Aggregation, signs of, 182. 
Ahmes, II, 54, 145. 
Alcuin, 16, 60, 61. 
Algebra 

in arithmetic, 16, 17, 68, 124, 130. 

ethical value of, 169. 

growth of, 145, 

kinds of, 155. 

name, ISI. 

practical value, 168. 

what, and why taught, 161, 165. 

when studied, 170. 
Al-Khowarazmi, I5I, 152, 201. 
Allman, 228 z. 
Al-Mamun, Ist. 
Al-Mansur, 150. 
Amusements of arithmetic, 15. 
Angle, 262, 274. 
Approximations, 142, 159. 
Arabic numerals, 50, 52, 53. 
Arabs, 5, I5I. 
Arbitrary value check, 190. 
Archimedes, 231, 238, 209. 
Argand, 213. 
Aristotle, 13, 47, 227. 
Arithmetic 


reasons for teaching, I, 19, 79, 98. 


history of teaching, 71. 
when to begin, 116. 
utilities of, 2, 7, 35. 
medizeval, 58. 
crystallizing, 64. 


Arithmetic 
oral, 117. 
commercial, 7, 136. 
first year of, 114. 
applied problems, 136. 
ancient divisions, 56. 
present status, 68. 
distinguished from algebra, 162, 
Arts, seven liberal, 4. 
Aryabhatta, 150. 
Ascham, 32. 
Assyrians, 5. 
Austrian methods, 122. 
Axioms, 178, 257, 262. 


Babylonians, 5, 50, 225. 

Bachet de Méziriac, 15. 

Bagdad, Ist. 

Bain, 24 2., 28. 

Ball, 241 7., 304. 

Beda, 7, 60. 

Beetz, 82 7. 

Beman, 148 %., 211 7. 

Beman and Smith 
arithmetic, 6672. 
algebra, 159 #. 
geometry, 285 2. 
trans. of Fink, 507., 304. 
trans. of Klein, 2907, 

Benedict, St., 60. 

Bertrand, 214. 

Bézout, 211. 

Biber, 80. 

Bibliography, 297. 

Biermann, 302, 


3°7 


308 


Blockmann, 80%. 
Boethius, 10, 59. 

Bologna, Io. 

Bolyai, 265. 
Boncompagni, 537. 
Boniface, St., 60. 

Bourlet, 163 7., 176, 219, 30%. 
Brahmagupta, 200. 
Brdutigam, 817. 
Bretschneider, 228 2. 
Brianchon, 232, 284. 
Brocard, 231. 

Brooks, 67 #., 299. 
Browning, 127. 

Biirgi, 67. 

Burnside and Panton, 301. 
Business arithmetic, 20. 
Busse, 58, 77. 


Cajori, 304. 

Calculi, 57. 

Calendar. See Easter, 61. 

Cantor, G., 106. 

Cantor, M., 117% 

Capella, 59. 

Cardan, 14, 153. 

Carnot, 232. 

Cassiodorus, 59. 

Catalan, 41. 

Cauchy, 169. 

Charlemagne, 60. 

Chasles, 228 7., 231, 232. 

Checks, 188. 

Chilperic, 59. 

Chinese, 2, 57. 

Chrystal, 163, 164 7., 176, 189, 216, 301. 

Chuquet, 153. 

Church schools, 5, 6, 15, 60, 62. 

Cicero, 6. 

Circle squaring, 290. 

Clairaut, 240. 

Clarke, 6. 

Cloister. See Church Schools. 

Colburn, 117. 

Comenius, 54. 

Committee of Ten, 69, 250, 281 7. 

Committee of Fifteen, 69, 70, 116. 

Compayré, 207., 847. 

Complex numbers. See Number sys- 
tems. 





INDEX 


Compound numbers, 22. 

Comte, 162, 186, 244. 

Conant, 442. 

Concentric circle plan, 88. 

Confucius, 33 2. 

Conrad, 14. 

Converse theorems, 277. 

Correlation, 3. 

Counting, 45. 

Court schools, 59. 

Cube, duplication of, 290. 

Culture value, 12, 20, 23, 27, 34, 39, 237, 
238. 

Cycloid, area of, 244. 


D'Alembert, 163, 220, 266. 
Date line, 129. 

Dauge, 163 7., 305. 
Davidson, 13%. 

Decimals. See Fractions. 
Definitions, 28, 176, 257. 

De Garmo, IIo, 111 %., 298. 
Desree, 177, 225. 

De Guin, 80. 

Delbos, 47. 

De Morgan, 44, 148 2., 177 2., 232. 
Denominate numbers, 37, 65. 
Denzel, 88. 

Desargues, 231. 

Descartes, 231. 

De Staél, 170. 

De Tilly, 266 2, 

Dewey, 45 7., 105, 299. 
Diesterweg, 18, 89. 
Diophantine equations, 150, 
Diophantus, 148. 

Discount, true, 35. 
Discovery, method of, 88. 
Dittes, 67. 

Division, 122, 

Dixon, 257 #. 

Dodgson, 229 #. 

Drawing, 241, 245, 271. 
Dressler, 1207. 

Duhamel, 29 7., 304. 
Duplication of the cube, 290, 
Dupuis, 303. 


Easter problem, 5, 7, 62. 
Ebers, Io, 


INDEX 


Egyptians, 10, II, 12, 50, 145, 226. 
Elimination, 211. 
Encyklopadie d. math. Wiss., 29 7. 
Equation 
in arithmetic, 16, 17, 68, 69, 124, 130. 
of payments, 65. 
classification of, 152. 
roots of numerical, 159. 
quadratic, 198. 
equivalent, 203. 
radical, 206. 
simultaneous, 208. 
diophantine, I50. 
Erfndungsmethode, 88, 
Euclid, 229, 235-238. 
Examinations, Io, 216, 
Exchange, 36, 65. 
Explanations, 140. 


Factor, 179. 
Factoring, 192, 197. 
Fahrmann, 462. 
Faifofer, 304. 
False position, 124. 
Fermat, 41. 
Ferrari, 154. 
Ferro, 14, 154. 
Fibonacci, 53. 
Fine, 186 7., 301. 
Fingers, 47, 58, IoI. 
Fink, 502., 304. 
Fiore, 14, 154. 
Fischer, 202. 
Fisher and Schwatt, 176, 
Fitch, 207., 24. 
Fitzga, 207. 
Formal solutions, 123. 
Formal steps, III. 
Fractions, II, 23, 54, I19. 
decimal, 55, 66, 119. 
Francke Institute, 75. 
Frisius, 14, 1007. 
Functions, 162, 163. 


Galileo, 244. 

Galley method, 67. 

Gaultier, 232. 

Gauss, 158, 213. 

Gemma Frisius, 14, 1007”, 
Generalization of figures, 279. 


3-2 


Geometry 
history of, 224. 
non-Euclidean, 233, 265, 269. 
defined, 234. 
limits, 236. 
why studied, 237. 
in the grades, 239, 243, 
demonstrative, 250, 27%. 
bases of, 257. 
impossible in, 289. 
solid, 290. 
inventional, 245, 
Gergonne, 232. 
Germain, 208. 
Gillespie, 162 z, 
Girard, 62. 
Girard, Pére, 83. 
Goldbach, 41. 
Goodwin, Bp., 171. 
Goschen, 176, 302. 
Gow, I1%., 227. 
Graffenried, 65 7. 
Grammateus, 63. 
Graphs, 208. 
Grass, 97 2. 
Grassmann, 106, 
Greatest common divisor, 39. 
Greeks, 6, 12, 50; 51, 55, 150, 227. 
Greenwood, 1252. 
Grube, 89, 118, 298. 
Grunert, 202. 
Guizot, 617. 


Hadamard, 285, 296, 303. 
Hall, G.S., 140. 

Hall and Stevens, 302. 
Halliwell, 53 2. 
Hamilton, 95. 

Hankel, 106, 225 2. 
Hanseatic League, 8, 62, 
Hanus, 139 #., 244, 347. 
Harms, 92 7., 245 2. 
Harpedonaptae, 226. 
Harpur Euclid, 302. 
Harriot, 156 2. 

Harris, 125 7. 
Harun-al-Raschid, 151. 
Hau computation, 145. 
Heath, 148 z. 

Hebrews, 50. 


310 


Heiberg, 263. 

Henrici, O., 189, 219, 237, 284 %., 303. 

Henrici and Treutlein, 284 2,, 303. 

Henry, 532. 

Hentschel, 89, 113, 114%. 

Heppel, 1907. 

Herbart, 95, III, 

Herodotus, 227, 

Heron, 148. 

Hilbert, 257 7., 266, 

Hill, 19 x. 

Hindu numerals, 50, 52, 53. 

History of mathematics, I, 42, 145, 
224. 

Holzmiiller, 173, 174, 251. 

Homogeneity as a check, IgI. 

Hoose, 85 2., 298. 

Horner, 160. 

Hotiel, 229, 242. 

Hiibsch, 16. 

Hudson, 166, 167 2., 170. 

Humanism, 13. 

Humbert, 299. 


Imaginaries. 
India, 3. 
Induction, 244. 

Interest, awakening, 220. 
simple and compound, 36. 
Interpretation of solutions, 220, 
Inventional geometry, 245. 

Involution, 31. 
Isidore, 60. 


See Number systems. 


Janicke, 75, 85 2. 
Janicke and Schurig, 722. 


Jews, 5. 
Journal Royal Asiatic Society, 527. 


Kallas, 92%. 

Kant, 95, 265. 

Kaselitz, 92. 

Kawerau, 87. 

Kehr, 72 7., 300. 

Kepler, 667, 

Khayyam, 201. 
Khowarazmi, 151, 152, 201, 
Klein, 265, 290 2, 

Klotzsch, 114”. 

Knilling, 207., 84, 867., 92, 94, 30I. 


INDEX 


Kéobel, 54. 
KG6nigsberger, 257 %. 
Kénnecke, 532%. 
Koreans, 57. 
Korner, 98 2. 
Kranckes, 88. 
Kriisi, 81. 


Laboratory methods, 76. 

Lacroix, 240. 

Laisant, II, 29%., 39, 49, Io4, 140, 156, 
240, 305. 

Lange, 96. 

Langley, 245. 

Laplace, 235. 

Laurie, 37. 

Lazzeri and Bassani, 304. 

Lemoine, 231. 

Leonardo Fibonacci of Pisa, 53. 

Liberal arts, seven, 4. 

Lobachevsky, 265. 

Loci, 282. 

Itacke, 31: 

Lodge, 125. 

Logarithms, 67. 

Logic in mathematics, 24, 25, 167, 207, 
238, 239. 

Logistics, 56. 

Longitude and time, 34, 126, 

Loria, 229. 

Lucas, 299. 


Macé, 32. 
Mahaffy, 13 2. 
Mahler, 304. 
Mamun, IsI. 
Mansur, I50. 
Martin, 6. 
Mathews, 239%. 
Matthiessen, 150 %., 203, 302. 
McClelland, 303. 
McCormack, 1762. 
McLellan and Dewey, 45 ., 105, 299. 
McMurry, 110, 298. 
Mensuration, 137. 
Mental gymnastic, 79, 84, 
Method, rise of, 74. 

great question of, Io9, 

in geometry, 283. 
Metric system, 134. 


INDEX 


Méziriac, 15. 

Middle ages, 58. 

Minchin, 251, 271. 

Minus and plus, 187. 

Mobius, 232. 

Mohammed ben Musa, I5I, 152, 201. 
Mohammedans, 4. 

Miiller, 45 2. 

Multiplication and division, 67, 74. 
Murhard, 172. 


Napier, 67. 
Negative numbers. 
tems. 
Neuberg, 231. 
Newcomb, 260%”, 
Newton, 48. 
Nixon, 302. 
Non-Euclidean geometry, 233. 
Notation, 48, 49, 112. 
Number systems, 157, 184, 213. 
concept, 99. 
pictures, 77. 


See Number sys- 


Object teaching, 71, Ioo, Io2, 

Obsolete in arithmetic, 68, 69, 70. 

Odd numbers, 57. 

Oliver, Wait, and Jones, 176, 

Omar Khayyam, 201. 

One-to-one correspondence, 106, 113, 
288, 295. 

Oral arithmetic, 117. 

Oriental algebra, 150. 

Oughtred, 156. 

Oxford, 9. 


Paolis, 304. 

Paris, University, 10. 

Paros, 92. 

Partnership, 65. 

Pascal, 259 ”, 

Payne’s translations, 20#., 847. 
Perception, 78. 

Pestalozzi, 18, 48, 58, 78, 116, 
Petersen, 285, 302. 
Philanthropin, 76. 

Phillips, 93 #., 298. 

Wy 2552250. 

Pincherle, 176, 219. 

Pitiscus, 662. 


311 


Pius II, 13. 

Plato, 12, 227, 229, 235. 
Pliicker, 232. 

Plus and minus, 187. 
Poincaré, 257 7. 
Poinsot, 163. 
Poncelet, 232. 

Postulates, 257, 262. 
Problems, statement of, 181. 
applied, in algebra, 215. 

Problem solvers, 14. 
Proklos, 233. 

Proportion, 36, 39, 129, 287. 
Puzzles, 40, 61. 

Pythagoras, 13, 228. 


Quadratic equations, 198. 
Quadrivium, 60. 


Rashdall, 5 2. 
Ratio idea of number, 48, 103. 
and proportion, 129, 287. 

Reasons for teaching mathematics, 1, 
I2, 17, 20, 23, 27, 34, 39, 237; 
238. 

Rebiére, 1267. 

Rechenmeister, 9, 63. 

Rechenschule, 62. 

Reciprocal theorems, 275. 

Recorde, 16, 1562. 

Reidt, 327. 

Rein, III, 247. 

Rein, Pickel, and Scheller, 24 #., 300. 

Remainder theorem, 195. 

Renaissance, 63. 

Reviews, 143. 

Rhyming arithmetics, 73. 

Riese, 14. 

Rochow, 77. 

Roman numerals, 50, 51, 54, 55. 

Rome, 6. 

Roots, 31. 

Rope stretchers, 226, 

Rosen, 1527. 

Rouché and De Comberousse, 285, 
303. 

Rousseau, 240. 

Rudolff, 63. 

Riiefli, 867. 

Ruhsam, 118. 


312 


Rules, 31, 72, 130, 167. 
Russell, 257 #. 


Saccheri, 233. 

Safford, 124 2. 

Sammlung Géschen, 176. 
Schubert, 176, 

Scales of counting, 46. 

Schafer, 79. 

Schiller, 238. 

Schmid, K. A., 2”. 

Schmidt, K., 6%. 

Schmidt, Z., 9. 

School World, 239 %., 252 2. 

Schotten, 2607. 

Schubert, 176, 186%., 265 7., 302. 

Schiiller, 302. 

Schurig, 300. 

Schuster, 124 7., 245%. 

Schwatt, 176, 239%. 

Scratch method, 67. 

Semites, 5. 

Servois, 232. 

Shaw, 245. 

Short cuts, 137. 

Signs. See Symbols. 

Similar figures, 261. 


Smith, D. E., 50z., 66., 158 2., 159 #., 


285 2., 290 2., 304. 
Socci and Tolomei, 304. 
Socrates, 6. 

Solon, 12. 

Spartans, 6. 

Speer, 103 7., 298. 
Spencer, W. G., 245. 
Spencer, H., 27 2. 
Spiral method, 118. 
Square root, 31. 

St. Benedict, 60. 

St. Boniface, 60. 
Stackel and Engel, 264. 
Stammer, 207. 
Standard time, 129. 
Staudt, 232. 

Stehn, 13, 147. 
Steiner, 232. 
Sterner, 67., 300. 
Stevin, 66. 

Straight line, 258. 
Sturm, 9. 


INDEX 


Subtraction, 121. 

Sully, 317. 

Surd, 180. 

Swan pan, 57. 

Sylvius, 13. 

Symbols, 66, 148, 155, 182, 222, 273, 
Symmetry as a check, Ig1. 


Tacitus, 58, 59. 

Tanck, 92, 94. 

Tannery, 162 7., 299. 
Tartaglia, 14, 154. 

Teachers’ failures, 26, 
Text-books, 70, 139, 173, 254. 
Thales, 227, 228. 

Theon of Alexandria, 131”. 
Tillich, 31, 77, 82, 86. 

Time, 34, 126. 

Tradition, Io. 

Trapp, 76. 

Trigonometry in algebra, 202, 
True discount, 35. 


Tiirk, 87. 

Twelve as a radix, 48. 
Tylor, 45 7. 

Unger, 7 7., 300. 


Universities, 8, 9. 
Utilities of arithmetic, 2, 20, 39. 


Veronese, 257 2. 
Vienna, Io, 

Vieta, 156, 201. 
Voltaire, 240. 

Von Busse, 58, 77. 
Von Rochow, 77. 
Von Staudt, 232. 


Wagner, 63. 
Walker, 39 #., 116. 
Wallis, 156 2. 
Ward, 42 7. 
Weber, 302. 
Weierstrass, 106, 
Wessel, 158, 213. 
Wordsworth, 51 2. 


Young, 174 2. 


Zahlenbilder, 77. 








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